Basic concepts in the geometry of Banach spaces

William B. Johnson^{1}

Department of Mathematics, Texas A&M University, College Station, Texas, USA

Joram Lindenstrauss^{2}

Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel

1 Supported in part by NSF DMS-9623260 and DMS-9900185, Texas Advanced Research Program 010366-163, and U.S.-Israel Binational Science Foundation. 2 Supported in part by U.S.-Israel Binational Science Foundation and by NSF DMS-9623260 as a participant in the workshop in linear analysis & probability at Texas A&M University.

Preprint submitted to Elsevier Preprint 7 February 2001

Contents

1 Introduction 3 2 Notations and special Banach spaces 3 3 Bases 7 4 Classical spaces 16 5 Banach lattices 25 6 Geometry of the norm 35 7 Analysis in Banach spaces 42 8 Finite dimensional Banach spaces 51 9 Local structure of infinite dimensional spaces 62 10 Some special classes of operators 72 11 Interpolation 88 12 List of symbols 97 References 99 1 Introduction

In this introductory chapter we present results and concepts which are often used in Banach space theory and will be used in articles in this Handbook without further reference. The material we treat, while familiar to experts in Banach space theory, has not made its way into introductory courses in functional analysis. The main purpose of this article is to make the subsequent articles accessible to anyone whose background includes basic graduate courses in analysis and functional analysis. Each section of this article is devoted to one aspect of Banach space theory.

Although this article can in no way be considered as an introductory course in Banach space theory, we do include indications of proof of some basic results in the hope that this will help the reader understand and get a feeling for the various concepts which are discussed. We reference only some of the (mostly introductory) books which treat the basic material we describe. Original sources are referenced in these books. We also mention some results which are not yet in elementary books on Banach spaces but which help to clarify the general picture. These generally were either discovered recently or are more difficult than the rest of the material. In these cases we refer to specific articles in this Handbook which treat the topic.

In general, we do not attach names to theorems (except when experts generally refer to the theorem with a name attached, such as the Hahn-Banach theorem, the Krein-Milman theorem, Rosenthal’s ℓ_{1} theorem,...) or give any historical background. Instead we refer to the books in the references as well as the articles in this Handbook.

2 Notations and special Banach spaces

Banach spaces will have either real or complex scalars. When the scalar field matters (for example, in results involving spectral theory or in theorems of an isometric nature or when analyticity plays a rôle), the scalar field is mentioned explicitly, but in the notation for special spaces the scalars are not specified.

Operators between Banach spaces are bounded and linear. An invertible operator T is called an isomorphism. Two norms on a vector space are called equivalent if the identity operator on X (with the two given norms) is an isomorphism. If ∥T ∥ = 1 = ∥T^{−1}∥, T is called an isometric isomorphism or simply an isometry and the domain and range of T are said to be isometric. We write X ≈ Y to denote that the spaces are isomorphic. To denote isometry we use the equal sign. An isomorphism from a Banach space onto itself is called an automorphism. A Banach space Y is said to be a quotient of the Banach space X if Y is isometric to X/Z for some closed subspace Z of X. By the open mapping theorem, Y is isomorphic to a quotient of X if there is an operator from X onto Y.

If X ≈ Y, d(X, Y) denotes the Banach-Mazur distance between the spaces, defined to be the infimum of ∥T ∥∥T^{−1}∥ as T ranges over all isomorphisms from X onto Y. So d(X, Y) = 1 if X and Y are isometric; the converse is true for finite dimensional spaces but not for infinite dimensional spaces. Note that the “triangle inequality” for the Banach-Mazur distance is submultiplicative rather than subadditive; that is, d(X, Y) ≤ d(X, Z)d(Z, Y).

A projection is an idempotent operator. A subspace Z of X is said to be complemented if there is a projection from X onto Z. This is the case if and only if Z is closed and there is a closed subspace W of X so that W ∩Z = {0} and X = W + Z; we then write X = W ⊕ Z and say that X is the direct sum of W and Z. In this case W is isomorphic to X/Z by the open mapping theorem.

We regard X as a subspace of X^{∗∗} under the canonical embedding. Note that there is always a projection of norm one from X^{∗∗∗} onto X^{∗} (restrict the functionals on X^{∗∗} to X).

An operator T : X → Y between Banach spaces is compact; respectively, weakly compact; if the image T B_{X} of the unit ball B_{X} of X has compact; respectively, weakly compact; closure in Y. If either X or Y is reflexive, then every operator from X to Y is weakly compact. The identity operator on X is compact if and only if X is locally compact if and only if X is finite dimensional.

A sequence {x_{n}}_{n=1}^{∞} in a Banach space X is called weakly Cauchy provided {x^{∗}(x_{n})}_{n=1}^{∞} converges for every x^{∗} in X^{∗}. Identifying a Banach space with a subspace of its bidual, and taking into account that the uniform boundedness principle implies that a weakly Cauchy sequence is bounded, we see that {x_{n}}_{n=1}^{∞} is weakly Cauchy in X if and only if it converges weak^{∗} in X^{∗∗}. The space X is said to be weakly sequentially complete provided every weakly Cauchy sequence {x_{n}}_{n=1}^{∞} in X converges weakly, which is the same as saying that the weak^{∗} limit of {x_{n}}_{n=1}^{∞} in X^{∗∗} is in X itself.

Here is a list of special classical Banach spaces and other objects. The elementary theory of these can be found in beginning texts in real and functional analysis. A more detailed list of symbols, including some notation undefined in the text because we regard it as “standard”, is contained in section 12.

ℕ = The natural numbers.

ℝ = The real numbers. ℂ = The complex numbers. 𝕋 = The unit circle in the complex plane. C(K; X) = Continuous functions f on the (usually) compact Hausdorff space

K taking values in the (usually) Banach space X, normed by ∥f∥ = sup_{t∈K} ∥f(t)∥.

C(K) = C(K; X) when X is the scalar field. L_{p}(μ) = The μ-measurable functions f for which ∥f∥_{p} = (∫ |f|^{p} dμ)^{1/p} < ∞.

Here 0 < p < ∞. L_{∞}(μ) = The μ-measurable essentially bounded functions, with norm ∥f∥_{∞} =

inf_{μA=0} sup |f_{|Ã}|. L_{p}(0, 1) = L_{p}(μ) when μ is Lebesgue measure on the unit interval. L_{p}(𝕋) = L_{p}(μ) when μ is normalized Lebesgue measure on the unit circle. ℓ_{p}(Γ) = L_{p}(μ) when μ is counting measure on the set Γ. ℓ_{p} = ℓ_{p}(Γ) when Γ = ℕ. ℓ_{p}^{n} = ℓ_{p}(Γ) when Γ = {1, 2, …, n}. c = The subspace of ℓ_{∞} of scalar sequences which have a limit. c_{0}(Γ) = The closure in ℓ_{∞}(Γ) of the scalar sequences which have finite sup-

port. c_{0} = c_{0}(Γ) when Γ = ℕ. B_{X} = The closed unit ball of the Banach space X. B_{X}(x, r) = The closed ball of radius r with center x in the Banach space X;

denoted also B(x, r) when X is understood. When 0 < p < 1, the space L_{p}(μ) is not a Banach space except in the trivial cases that it is zero or one dimensional. The metric on L_{p}(μ), 0 < p < 1, is given by ρ(x, y) = ∥x − y∥^{p}. If {X_{n}}_{n=1}^{∞} is a sequence of Banach spaces and 1 ≤ p ≤ ∞, (∑ X_{n})_{p} is the Banach space of all sequences {x_{n}}_{n=1}^{∞} with x_{n} ∈ X_{n} and ∥{x_{n}}_{n=1}^{∞}∥ := ∥{∥x_{n}∥}_{n=1}^{∞}∥_{p} < ∞. (∑ X_{n})_{0} is the closed subspace of (∑ X_{n})_{∞} for which ∥x_{n}∥ → 0. When all X_{n} are the same space X we sometimes write ℓ_{p}(X) instead of (∑ X)_{p}. While this standard notation conflicts with the (also standard) notation ℓ_{p}(Γ) introduced above, no confusion should arise. Sometimes also uncountable direct sums will be considered (with the obvious meaning). The elementary duality relations c_{0}^{∗} = ℓ_{1} and ℓ_{p}^{∗} = ℓ_{q}, 1 ≤ p < ∞, \frac{1}{p} + \frac{1}{q} = 1, yield that (∑ X_{n})_{0}^{∗} = (∑ X_{n}^{∗})_{1} and (∑ X_{n})_{p}^{∗} = (∑ X_{n}^{∗})_{q} for 1 ≤ p < ∞. When the product X ⊕ Y (called the direct sum of X and Y) is given the norm ∥(x, y)∥ = ∥(∥x∥, ∥y∥)∥_{p}, we write the resulting space X ⊕_{p} Y. All these norms on X ⊕ Y are equivalent. When we are interested in X ⊕ Y only up to isomorphism we do not specify a particular norm.

Probability theory has had a profound impact on Banach space theory. Consequently, we shall use in places probabilistic terminology. For example, we call the function which is one on a set A and zero on the complement of A the indicator function and denote it by 1_{A}. We recall now some other standard concepts and results in probability theory which can be found in most introductory textbooks in the subject; in particular, in [10]. A random variable is a real valued measurable function on a probability space. A random variable g is said to be Gaussian or to have Gaussian distribution or normal distribution if its distribution function ℙ[g ≤ t] is equal to \frac{1}{σ\sqrt{2π}} ∫_{−∞}^{t} e^{\frac{−(x−a)^{2}}{2σ^{2}}} dx. Here ℙ denotes the underlying probability measure. The constant a is the mean or expectation or expected value of g, defined by 𝔼g = ∫ g dℙ. The quantity σ^{2} is the variance of g, defined to be 𝔼(g − a)^{2}. If a = 0 and σ = 1, g is called a standard Gaussian variable.

A finite collection {f_{n}}_{n=1}^{N} of random variables on the same probability space is called independent provided that ℙ ∩_{n=1}^{N} [f_{n} ∈ A_{n}] = ∏_{n=1}^{N} ℙ[f_{n} ∈ A_{n}] for all Borel sets A_{n}. So 𝔼(f · g) = 𝔼f𝔼g if f and g are independent. An arbitrary collection of random variables is independent provided each finite subcollection of the collection is independent. Given a sequence {f_{n}}_{n=1}^{∞} of random variables with f_{n} defined on the probability space (Ω_{n}, ℙ_{n}), one can construct a sequence {g_{n}}_{n=1}^{∞} of independent random variables so that g_{n} has for each n the same distribution as f_{n}. Indeed, let (Ω, ℙ) be the infinite product of the probability spaces (Ω_{n}, ℙ_{n}) and for ω = {ω_{n}}_{n=1}^{∞} in Ω set g_{n}(ω) = f_{n}(ω_{n}). In particular, it is possible to define a sequence of independent random variables having standard Gaussian distribution on the infinite product (0, 1)^{ℕ} of (0, 1) (with the product of Lebesgue measure). By the isomorphism theorem for separable measure algebras ([18, p. 399]), (0, 1)^{ℕ} can be replaced by (0, 1) itself, but it is often more convenient to work on the product space.

The characteristic function of the random variable g is the function φ : ℝ → ℂ defined by φ(t) = 𝔼e^{itg}. Useful algebraic identities include φ(−t) = \overline{φ(t)}; φ_{ag+b}(t) = e^{ibt}φ(at); and, especially, φ_{f+g} = φ_{f}φ_{g}, valid when f and g are independent. The characteristic function of a standard Gaussian random variable is e^{− \frac{t^{2}}{2}} .

A key fact is that two random variables (possibly defined on different probability spaces) have the same distribution if and only if they have the same characteristic function. This is proved by an inversion formula [10, 2.3.a], another simple consequence of which is that the characteristic function of a random variable g is real valued if and only if g is symmetric; that is, g and −g have the same distribution. A (by no means immediate) consequence of the inversion formula [10, 2.7] is that for 0 < r < 2 the function e^{−|t|^{r}} is the characteristic function of a (necessarily symmetric) random variable, called a symmetric r-stable random variable. The tail distribution ℙ[|g| > t] of a symmetric r-stable random variable g is like t^{−r} as t → ∞, so that ∥g∥_{p} < ∞ for p < r, but ∥g∥_{r} = ∞.

Another probabilistic notion that plays an important role in Banach space theory is that of martingale. First we recall the notion of conditional expectation. Let ℙ be a probability measure on a σ-algebra B. If A is a sub σ-algebra of B and f is a ℙ-integrable function, then by the Radon-Nikodým theorem there is a A-measurable function g so that for each A in A, ∫_{A} f dℙ = ∫_{A} g dℙ. The function g is called the conditional expectation of f given A and is sometimes denoted by 𝔼(f|A). Suppose now that {f_{n}}_{n=0}^{∞} is a sequence of random variables on the same probability space and B_{n} is the smallest σ-algebra for which f_{0}, f_{1}, …, f_{n} are measurable. The sequence {f_{n}}_{n=0}^{∞} is called a martingale provided that for each n, f_{n} = 𝔼(f_{n+1}|B_{n}). The sequence {d_{n}}_{n=0}^{∞} of differences; d_{0} = f_{0}, d_{n} = f_{n} − f_{n−1}; is then called a martingale difference sequence. Notice that a sequence {d_{n}}_{n=0}^{∞} of independent random variables is a martingale difference sequence if and only if d_{n} has mean zero for all n ≥ 1. A simple but important example of a martingale difference sequence which is not a sequence of independent random variables is the Haar system, discussed in section 4. One basic theorem about martingales, the martingale convergence theorem (see [10, 4.2.10]), states that every L_{1} bounded martingale converges a.e., which means that if {f_{n}}_{n=0}^{∞} is a martingale of ℙ-measurable functions and sup_{n} 𝔼|f_{n}| < ∞, then {f_{n}}_{n=0}^{∞} converges a.e. Moreover, if the martingale is uniformly integrable (see section 4, then it also converges in L_{1}(ℙ) (see [10, 4.5.3]).

3 Bases

Excepting [2] and [12], which are oriented to Banach space theory, few introductory texts in functional analysis treat Schauder bases. Nevertheless, bases are a very useful tool for investigating properties of Banach spaces.

We prove few statements in this section since most of the results are proved in [2] and [12]. The book [14] often contains only sketches of proofs. Chapter 2 of [21] contains enough details to be pleasant reading for the mature student or experienced mathematician who is not an expert in Banach space theory. All of these books contain exercises. Many in [21] are challenging even after peeking at the hints. Exercises in [14], as in this introductory article, are scattered throughout the text and are sometimes flagged by such expressions as “clearly”, “hence”, and the like.

A Schauder basis or simply a basis for a Banach space X is a sequence {x_{n}}_{n=1}^{∞} of vectors in X such that every vector in X has a unique representation of the form ∑ α_{n}x_{n} with each α_{n} a scalar and where the sum is converges in the norm topology. The mapping x ↦ α_{n} then defines for each n a linear functional x_{n}^{∗} on X. One checks that the expression !x! = sup_{n} ∥ ∑_{k=1}^{n} x_{k}^{∗}(x)x_{k}∥ defines a stronger complete norm on X, so that !·! and ∥ · ∥ are equivalent by the open mapping theorem. One deduces from this that the biorthogonal functionals for a basis are necessarily continuous. Moreover, the biorthogonal functionals are a basic sequence in X^{∗}; that is, they form a basis for their closed linear span. When it is useful to specify the biorthogonal functionals, we sometimes refer to the “basis” {x_{n}, x_{n}^{∗}}_{n=1}^{∞}.

A sequence {x_{n}}_{n=1}^{∞} is called normalized if each vector x_{n} has norm one and {x_{n}}_{n=1}^{∞} is called seminormalized if it is bounded and bounded away from zero. For most purposes it is sufficient to consider normalized or at least seminormalized basic sequences.

The partial sum projections of a basis {x_{n}, x_{n}^{∗}}_{n=1}^{∞}, defined for n = 1, 2, … by P_{n}x = ∑_{i=1}^{n} x_{i}^{∗}(x)x_{i}, are necessarily uniformly bounded and converge strongly to the identity; that is, ∥x − P_{n}x∥ → 0 for each x in X. The supremum of the norms of these partial sum projections is called the basis constant. This quantitative notion is of interest also if we just consider finite basic sequences or bases for finite dimensional spaces. A sequence {x_{n}}_{n=1}^{∞} of nonzero vectors is basic with basis constant at most C if and only if for all n < m (and all scalars α_{i}) the inequality ∥ ∑_{i=1}^{n} α_{i}x_{i}∥ ≤ C∥ ∑_{i=1}^{m} α_{i}x_{i}∥ holds. A block basis {y_{j}}_{j=1}^{∞} of the basis {x_{n}}_{n=1}^{∞} is a sequence of nonzero vectors of the form

y_{j} = ∑_{k=n_{j}+1}^{n_{j+1}} α_{k}x_{k} for some sequence n_{1} < n_{2} < . . .. The basis constant of a

block basis of {x_{n}}_{n=1}^{∞} is no larger than the basis constant of {x_{n}}_{n=1}^{∞}.

A basis is called monotone provided that its basis constant is one. One can change to an equivalent norm, |||·|||, in which the basis is monotone: just set |||x||| = sup_{n} ∥P_{n}x∥. In fact, if one defines instead |||x||| = sup_{n<m} ∥P_{m}x−P_{n}x∥ one gets an equivalent norm which is even bimonotone; that is, the basis is monotone and also all the complementary projections I − P_{n} have norm one.

Often one constructs a basic sequence {x_{n}}_{n=1}^{∞} by induction and checks that it

is basic by verifying for each n the inequality ∥ ∑_{i=1}^{n} α_{i}x_{i}∥ ≤ C_{n}∥ ∑_{i=1}^{n+1} α_{i}x_{i}∥ with

∑_{n=1}^{∞}(C_{n}−1) < ∞; the basis constant of {x_{n}}_{n=1}^{∞} is then at most ∏_{n=1}^{∞} C_{n}. Here is one classical construction, which yields, for example, that every infinite dimensional Banach space contains a basic sequence. Given any sequence C_{n} as above and having selected x_{1}, …, x_{n}, take a finite set S = S_{n} of norm one linear functionals on X which almost determine the norm on the linear span X_{n} of x_{1}, …, x_{n} in the sense that for each x ∈ X_{n}, ∥x∥ ≤ C_{n} max_{x^{∗}∈S} |x^{∗}(x)|. Now x_{n+1} can be any nonzero vector in the finite codimensional subspace S_{⊥} of X. This construction allows considerable “play”–given ϵ > 0, as long as you select x_{n+1} so that max_{x^{∗}∈S} ∥x_{n+1}∥^{−1}|x^{∗}(x_{n+1})| is sufficiently small, the resulting sequence {x_{n}}_{n=1}^{∞} will have basis constant less than ϵ+∏_{n=1}^{∞} C_{n}. This yields, for example, that every weakly null, non-norm null sequence has a basic subsequence.

This last result can be improved substantially. Two basic sequences {x_{n}}_{n=1}^{∞} and {y_{n}}_{n=1}^{∞} are equivalent provided that the map T x_{n} = y_{n} extends to an isomorphism from the closed span of {x_{n}}_{n=1}^{∞} onto the closed span of {y_{n}}_{n=1}^{∞}; K-equivalent if ∥T ∥∥T^{−1}∥ ≤ K (so, strangely, {x_{n}}_{n=1}^{∞} is 1-equivalent to {2x_{n}}_{n=1}^{∞}, but usually basic sequences are normalized). The principle of small perturbations says that if {x_{n}}_{n=1}^{∞} is a basic sequence in X and ∥x_{n} − y_{n}∥ → 0 sufficiently quickly, then {y_{n}}_{n=1}^{∞} is a basic sequence which is equivalent to {x_{n}}_{n=1}^{∞}. To prove this, let {x_{n}^{∗}}_{n=1}^{∞} be Hahn-Banach extensions of the functionals biorthogonal to {x_{n}}_{n=1}^{∞} to functionals in X^{∗} and define an operator S on X by Sx = ∑ x_{n}^{∗}(x)(y_{n} − x_{n}), so ∥S∥ ≤ ∑ ∥x_{n}^{∗}∥∥y_{n} − x_{n}∥. If ∥S∥ < 1, elementary considerations yield that I + S is an automorphism of X which maps x_{n} to y_{n}. This also yields that if {x_{n}}_{n=1}^{∞} is a basis of X, then so is {y_{n}}_{n=1}^{∞}. From this one sees that if {x_{n}}_{n=1}^{∞} is a basis for X with biorthogonal functionals {x_{n}^{∗}}_{n=1}^{∞}, if {y_{n}}_{n=1}^{∞} is a seminormalized sequence in X, and if for each k, x_{k}^{∗}(y_{n}) → 0 as n → ∞, then {y_{n}}_{n=1}^{∞} has a subsequence which is equivalent to some block basis {z_{n}}_{n=1}^{∞} of {x_{n}}_{n=1}^{∞}. From this it follows that every infinite dimensional subspace Y of a Banach space X with a basis {x_{n}}_{n=1}^{∞} contains, for every K > 1, a normalized basic sequence {z_{n}}_{n=1}^{∞} which is K-equivalent to a normalized block basis {y_{n}}_{n=1}^{∞} of {x_{n}}_{n=1}^{∞}. Moreover, the small perturbation argument indicated above shows that the generated isomorphism which maps {y_{n}}_{n=1}^{∞} into Y extends to an automorphism on X.

To see how these basic facts about bases might be used, suppose that T is a noncompact operator from a subspace X_{0} of a Banach space X with basis {x_{n}, x_{n}^{∗}}_{n=1}^{∞} into a Banach space Y with basis {z_{n}}_{n=1}^{∞}. Then there is a sequence {y_{n}}_{n=1}^{∞} in the unit ball of X_{0} such that for some ϵ > 0 and all n ≠ m, ∥T y_{n} − T y_{m}∥ > ϵ. By passing to a subsequence of differences of {y_{n}}_{n=1}^{∞} it can be assumed that for each k, x_{k}^{∗}(y_{n}) → 0 as n → ∞. In view of what was discussed in the previous paragraph, by passing to another subsequence it can be assumed that {y_{n}}_{n=1}^{∞} is an arbitrarily small perturbation of a block basis of {x_{n}}_{n=1}^{∞}. Repeating the same argument for {T y_{n}}_{n=1}^{∞} in Y and normalizing at the end, we conclude that if T is a noncompact operator from a subspace X_{0} of a Banach space X with basis {x_{n}}_{n=1}^{∞} into a Banach space Y with basis {z_{n}}_{n=1}^{∞}, then there are automorphisms U on X and V on Y and a normalized block basis {y_{n}}_{n=1}^{∞} of {x_{n}}_{n=1}^{∞} with y_{n} ∈ U^{−1}X_{0} and such that {V T Uy_{n}}_{n=1}^{∞} is a seminormalized block basis of {z_{n}}_{n=1}^{∞}.

This last mentioned result gives considerable information in cases where all block bases of a basis can be characterized. The simplest examples of bases are provided by the unit vector basis {e_{n}}_{n=1}^{∞} for ℓ_{p}, 1 ≤ p < ∞, and c_{0}. It is evident that in these spaces every normalized block basis of {e_{n}}_{n=1}^{∞} is 1-equivalent to {e_{n}}_{n=1}^{∞} and every seminormalized block basis of {e_{n}}_{n=1}^{∞} is equivalent to {e_{n}}_{n=1}^{∞}. Thus we can conclude that every operator from a subspace of ℓ_{r} into ℓ_{p}, 1 ≤ p < r < ∞, is compact. This shows in particular that no infinite dimensional subspace of ℓ_{r} isomorphically embeds into ℓ_{p} when 1 ≤ p, r < ∞ and p ≠ r (an isomorphism between infinite dimensional spaces cannot be compact since no infinite dimensional space is locally compact).

We should mention that, in contrast to the case of ℓ_{p} and ℓ_{r}, it generally is quite difficult to determine whether two spaces are isomorphic even when both are presented concretely as similar but different function spaces.

Another consequence of the principle of small perturbations is that ℓ_{1} has the Schur property, which means that every weakly convergent sequence in ℓ_{1} is norm convergent. Indeed, otherwise there would be a weakly null sequence {x_{n}}_{n=1}^{∞} in ℓ_{1} which is bounded away from zero. The sequence {x_{n}}_{n=1}^{∞} is necessarily bounded, so the sequence {x_{n}}_{n=1}^{∞} would have a subsequence {x_{n_{k}}}_{k=1}^{∞} which is equivalent to a block basis of the unit vector basis of ℓ_{1} and thus equivalent to the unit vector basis of ℓ_{1}. This is a contradiction because the unit vector basis of ℓ_{1} is not weakly null. An easy formal consequence of the fact that ℓ_{1} has the Schur property is that every weakly Cauchy sequence in ℓ_{1} is norm convergent. In particular, ℓ_{1} is weakly sequentially complete.

The most natural basis for L_{p}(0, 1), 1 ≤ p < ∞, is the Haar system {h_{n}}_{n=0}^{∞}, where h_{0} = 1_{[0,1)}, and for n = 2^{j} + k with j = 0, 1, … and k = 0, 1, …, 2^{j} − 1,

h_{n} = 1_{[k2^{−j},(2k+1)2^{−j−1})} − 1_{[(2k+1)2^{−j−1},(k+1)2^{−j})}.

It is easy to check that if 1 ≤ p ≤ ∞, then for each n and all scalars α_{i} the inequality ∥ ∑_{i=0}^{n} α_{i}h_{i}∥_{p} ≤ ∥ ∑_{i=0}^{n+1} α_{i}h_{i}∥_{p} holds, which means that the Haar system is a monotone basic sequence in L_{p}(0, 1). Since the linear span of the Haar system is dense in L_{p}(0, 1) when p < ∞, the Haar basis is a monotone basis for L_{p}(0, 1) when 1 ≤ p < ∞.

The sequence {f_{n}}_{n=0}^{∞}, where f_{0} = 1_{[0,1]} and for n ≥ 1, f_{n} is the indefinite integral of the Haar function h_{n−1}, forms a monotone basis for C[0, 1]. When normalized in the supremum norm, this is called the Faber-Schauder basis.

The trigonometric system {e^{inθ}}_{n=−∞}^{∞} forms a basis for L_{p}(𝕋) when it is ordered 0, 1, −1, 2, −2, … and 1 < p < ∞. This follows from the boundedness of the Riesz projection in this range. The Riesz projection is defined formally by R(f) = ∑_{n=0}^{∞} f̂(n)e^{inθ}, where f̂(n) = (2π)^{−1} ∫_{0}^{2π} f(t)e^{−int} dt is the nth Fourier coefficient of the function f. The unboundedness of the Riesz projection on L_{1}(𝕋) and C(𝕋) yields that the trigonometric system is not a basis for these spaces.

A series ∑ x_{n} in a Banach space is said to converge unconditionally provided every rearrangement of the series converges. This is equivalent to ∑ x_{k_{n}} converges for each subsequence of {x_{n}}_{n=1}^{∞} and also to ∑ ±x_{n} converges for each choice of signs ±. Except in finite dimensional spaces, unconditional convergence is weaker than absolute convergence; that is, convergence of ∑ ∥x_{n}∥ (see section 8).

A basis {x_{n}}_{n=1}^{∞} is said to be an unconditional basis provided that ∑ α_{n}x_{n} converges unconditionally whenever it converges. This is equivalent to saying that every permutation of {x_{n}}_{n=1}^{∞} is also a basis. If {x_{n}}_{n=1}^{∞} is an unconditional basis for the Banach space X and θ = {θ_{n}}_{n=1}^{∞} is a sequence of ±1’s, define S_{θ} : X → X by S_{θ}(∑ α_{n}x_{n}) = ∑ θ_{n}α_{n}x_{n}. The supremum over all such θ of ∥S_{θ}∥ is finite, and is called the unconditional constant of the basis. One can define an equivalent norm |||·||| on X for which the unconditional basis has unconditional constant one by setting |||x||| = sup_{θ} ∥S_{θ}x∥. Such a norm is said to be unconditionally monotone. When the norm is unconditionally monotone, all permutations of the basis are monotone (sometimes this weaker condition is referred to as “unconditionally monotone”).

A sequence which is an unconditional basis for its closed linear span is said to be unconditionally basic. Since a block basis of an unconditional basis is clearly unconditional (with unconditional constant no larger than that of the basis itself), the perturbation principle described above yields that every infinite dimensional subspace of a space with unconditional basis contains an unconditionally basic sequence.

The unit vector bases for ℓ_{p}, 1 ≤ p < ∞, and c_{0} are the simplest examples of unconditional bases. For 1 < p < ∞, the Haar system forms an unconditional basis for L_{p}(0, 1). The “modern” proof of this proceeds via martingale theory. The exact unconditional constant of the Haar system in L_{p}(0, 1) is computed in [23]. In section 8 we point out that the trigonometric system is an unconditional basis in L_{p}(𝕋) only for p = 2.

A basis {x_{n}}_{n=1}^{∞} for X is shrinking provided the linear span of the biorthogonal functionals {x_{n}^{∗}}_{n=1}^{∞} is (norm) dense in X^{∗}, which is to say that {x_{n}^{∗}}_{n=1}^{∞} is a basis for X^{∗}. Notice that a bounded shrinking basis necessarily converges weakly to zero. The basis {x_{n}}_{n=1}^{∞} is boundedly complete provided that whenever the sequence {∑_{i=1}^{n} α_{i}x_{i}}_{n=1}^{∞} is bounded, then it is convergent. If a basis is shrinking, then its biorthogonal functionals are a boundedly complete basis for X^{∗}. Conversely, if the biorthogonal functionals form a boundedly complete basis for their closed linear span Y, then the basis is shrinking (and hence Y = X^{∗}). Similarly, a basis is boundedly complete if and only if its biorthogonal functionals are a shrinking basis for their closed linear span. If {x_{n}}_{n=1}^{∞} is a boundedly complete basis for X and Y is the normed closed linear span of the biorthogonal functionals {x_{n}^{∗}}_{n=1}^{∞}, the shrinkingness of {x_{n}^{∗}}_{n=1}^{∞} implies that the natural evaluation mapping from X into Y^{∗} is a surjective isomorphism (which is easily seen to be an isometry if the basis is monotone). Consequently, a space with a boundedly complete basis is isomorphic to a separable conjugate space. From these facts it follows, in particular, that a Banach space X with a basis is reflexive if and only if some [or every] basis for X is both shrinking and boundedly complete.

If X has an unconditional basis, then some [or every] unconditional basis is boundedly complete if and only if X has no subspace isomorphic to c_{0}, while some [or every] unconditional basis is shrinking if and only if no [or no complemented] subspace of X is isomorphic to ℓ_{1}. Consequently, a space with unconditional basis is reflexive if and only if no subspace is isomorphic to either c_{0} or to ℓ_{1} if and only if its second dual is separable. In contrast, there is a nonreflexive space which has both a shrinking basis and a boundedly complete basis which is of codimension one in its bidual, [14, 1.d.2].

While most of the separable spaces encountered in classical analysis have bases, many do not have an unconditional basis; in particular, the spaces L_{1}(0, 1) and C[0, 1]. Indeed, c_{0} does not embed isomorphically into L_{1}(0, 1) (for example, because L_{1}(0, 1) is weakly sequentially complete and c_{0} is not; see section 4), so any unconditional basis for L_{1}(0, 1) would have to be boundedly complete and L_{1}(0, 1) would be isomorphic to a separable conjugate space. But L_{1}(0, 1) does not even embed isomorphically into a separable conjugate space (since if L_{1}(0, 1) ⊂ X^{∗} and for t ∈ (0, 1) x_{t}^{∗} is a weak^{∗} cluster point of the sequence n^{−1}1_{(t,t+1/n)}, then it can be checked that there is an uncountable subset S of (0, 1) and δ > 0 so that for all t ≠ s in S, ∥x_{t}^{∗} −x_{s}^{∗}∥ > δ). A more sophisticated (but actually technically easier) argument shows even that L_{1}(0, 1) does not embed isomorphically into a space with unconditional basis; see [14, 1.d.1]. Since, as mentioned in section 4, every separable space embeds isometrically into C[0, 1], C[0, 1] also does not embed isomorphically into a space with unconditional basis.

It is a deep result that there exist infinite dimensional spaces which do not contain an unconditionally basic sequence (see [37]). The example uses Banach spaces whose norm is not defined explicitly by a formula but by an implicit procedure. This method of defining a Banach space (or a norm) is very useful in many contexts.

A basis {x_{n}}_{n=1}^{∞} is called symmetric provided every permutation of {x_{n}}_{n=1}^{∞} is equivalent to {x_{n}}_{n=1}^{∞}. In particular, every permutation of {x_{n}}_{n=1}^{∞} is a basis, so a symmetric basis is unconditional. A basis {x_{n}}_{n=1}^{∞} is called subsymmetric provided it is unconditional and equivalent to each subsequence of itself. A symmetric basis is subsymmetric [14, 3.a.3]. It is evident that if a basis is unconditional, symmetric, or subsymmetric, then the same is true for the biorthogonal functionals (in their closed linear span).

The unit vector bases for ℓ_{p}, 1 ≤ p < ∞, and c_{0} are symmetric, while the L_{p}- normalization of the Haar system is not a symmetric basis for L_{p}(0, 1) if p ≠ 2. In fact, L_{p}(0, 1) has no subsymmetric basis if p ≠ 2. We have already mentioned that L_{1}(0, 1) does not have even an unconditional basis. That L_{p}(0, 1), 2 < p < ∞, has no subsymmetric basis follows from the dichotomy principle for sequences in the space, which is discussed in section 8. This principle says that if {x_{n}}_{n=1}^{∞} is a seminormalized basic sequence in L_{p}(0, 1), 2 < p < ∞, then {x_{n}}_{n=1}^{∞} has a subsequence which is equivalent to the unit vector basis for either ℓ_{p} or ℓ_{2}. Since, as noted in section 4, L_{p}(0, 1), p ≠ 2, is not isomorphic to either ℓ_{p} or ℓ_{2}, L_{p}(0, 1), 2 < p < ∞, has no subsymmetric basis. That L_{p}(0, 1), 1 < p < 2, has no subsymmetric basis then follows by duality.

It is interesting that symmetric bases are nevertheless prevalent. For example, any space which has an unconditional basis is isomorphic to a complemented subspace of a space which has a symmetric basis (this will be discussed further in section 11). In fact, there is even a space Y which has a symmetric basis and also an unconditional basis {z_{n}}_{n=1}^{∞} which is universal for unconditional bases in the sense that every unconditional basis for any Banach space is equivalent to a subsequence of {z_{n}}_{n=1}^{∞}! The space Y and the universal unconditional basis is easy to construct, given the fact, to be proved in section 4, that every separable Banach space is isometric to a subspace of C[0, 1]. Indeed, one takes a dense sequence {x_{n}}_{n=1}^{∞} of non zero vectors in C[0, 1] and lets Y be the completion of the finitely supported scalar sequences under the norm ∥{a_{n}}_{n=1}^{∞}∥ := sup_{±} ∥ ∑_{n} ±a_{n}x_{n}∥_{C[0,1]}. The unit vector basis in Y is a universal unconditional basis. See [14, p. 129] or [41] for further discussion and for the proof that the space Y has a symmetric basis.

While unconditional bases, symmetric bases, and subsymmetric bases are all useful strengthenings of the notion of basis, there are equally useful weakenings. A Schauder decomposition for a Banach space X is a sequence {X_{n}}_{n=1}^{∞} of nonzero closed subspaces of X such that every vector x in the space has a unique representation of the form x = ∑ x_{n} with x_{n} ∈ X_{n}. If each X_{n} is finite dimensional, the decomposition is called a finite dimensional decomposition, or simply an FDD. A basis can be regarded as an FDD {X_{n}}_{n=1}^{∞} such that every space X_{n} is one dimensional. As in the case of bases, a Schauder decomposition {X_{n}}_{n=1}^{∞} for a Banach space X determines a sequence {P_{n}}_{n=1}^{∞} (called the partial sum projections of the decomposition) of commuting projections with increasing ranges which converge strongly to the identity operator on X; namely, for x = ∑ x_{n} with x_{n} ∈ X_{n} and n = 1, 2, …, P_{n}x = ∑_{i=1}^{n} x_{i}. Conversely, if {P_{n}}_{n=1}^{∞} is a sequence of commuting projections with increasing ranges which converge strongly to the identity operator on X, then {(P_{n}−P_{n−1})X}_{n=1}^{∞} (P_{0} := 0) is a Schauder decomposition for X for which {P_{n}}_{n=1}^{∞} is the sequence of partial sum projections. The biorthogonal functionals for a basis are replaced in the case of a Schauder decomposition by the sequence {(P_{n}^{∗} − P_{n−1}^{∗})X^{∗}}_{n=1}^{∞} of (even weak^{∗}) closed subspaces of X^{∗}, which form a Schauder decomposition for their closed span. The supremum of the norms of the partial sum projections determined by a Schauder decomposition is finite and is called the decomposition constant of the decomposition, and the Schauder decomposition is called monotone if its decomposition constant is one.

The definitions of unconditional, shrinking, and boundedly complete bases generalize immediately to Schauder decompositions. The structure theory for bases goes over with little difficulty to FDD’s. More importantly, there are useful concepts involving FDD’s which are not merely generalizations of notions about bases. For example, a blocking of a Schauder decomposition {X_{n}}_{n=1}^{∞} is a sequence of the form {X_{n_{k}} + … + X_{n_{k+1}−1}}_{k=1}^{∞} with 1 = n_{1} < n_{2} < . . .. Any blocking of a basis (that is, of a Schauder decomposition into one dimensional spaces) produces an FDD which is not a basis. While bases and block bases are fine in situations where passage to subsequences or subspaces involves no loss, blockings of FDD’s sometime provide the proper framework for investigations where it is important to maintain global control. For example, it was mentioned above that every normalized basic sequence in ℓ_{p}, 1 < p < ∞, has a subsequence equivalent to the unit vector basis of ℓ_{p}, but one can prove that every FDD for a subspace of ℓ_{p}, 1 < p < ∞, has a blocking which is an ℓ_{p} decomposition [14, 2.d.1]. (A Schauder decomposition {X_{n}}_{n=1}^{∞} is said to be an ℓ_{p} decomposition provided that for x_{n} ∈ X_{n}, the series ∑ x_{n} converges if and only if ∑ ∥x_{n}∥^{p} < ∞).

A separable Banach space is said to have the bounded approximation property or simply BAP provided there is a sequence {T_{n}}_{n=1}^{∞} of finite rank operators on X which converges strongly to the identity. If the operators can be chosen to commute, the space has the commuting bounded approximation property (CBAP). In either case the supremum of ∥T_{n}∥ is finite; if at most λ, we say that the space has the λ-BAP or λ-CBAP, or when λ = 1 the metric approximation property (MAP) or commuting metric approximation property (CMAP). Finally, A Banach space X has the approximation property or simply AP provided that for every compact subset K of X and ϵ > 0, there is a finite rank operator T on X such that ∥x − T x∥ < ϵ for every x ∈ K. If a space X has the AP, then every compact operator S into X is the norm limit of a sequence of finite rank operators. Indeed, take for n = 1, 2, … a finite rank operator T_{n} on X so that for each x in the image under of the unit ball of the domain of S the inequality ∥x − T_{n}x∥ ≤ 1/n holds. Then ∥S − T_{n}S∥ ≤ 1/n. The converse is also true [14, 1.e.4].

The implications basis ⇒ FDD ⇒ CBAP ⇒ BAP ⇒ AP are all easy or obvious. None are reversible except possibly CBAP ⇒ BAP; this is discussed in [24], as is the existence of spaces which fail the AP and connections among these various approximation conditions. There are some surprises, such as the AP implies the MAP in reflexive spaces; the MAP implies CMAP; and a space which has the BAP is isomorphic to a complemented subspace of a space which has a basis. Incidentally, while spaces failing the AP or other approximation properties might be thought pathological, they arise naturally in various contexts (for example, the nonseparable space B(ℓ_{2}) of operators on ℓ_{2} fails the AP, as do some separable C^{∗} algebras). Moreover, the construction of spaces failing approximation conditions leads to nice theorems which do not involve any notion of approximation; in particular, isomorphic characterizations of Hilbert space (see [35]).

The existence of spaces failing the AP as well as other considerations lead to the study of other basis-like structures in Banach spaces. Probably the most useful of these is that of Markuschevich basis. A Markuschevich basis for a Banach space X is a biorthogonal system {x_{γ}, x_{γ}^{∗}}_{γ∈Γ} for which {x_{γ}}_{γ∈Γ} is fundamental (that is, the span of the x_{γ}’s is dense in X) and {x_{γ}^{∗}}_{γ∈Γ} is total; that is, the x_{γ}^{∗}’s separate the points of X. The trigonometric system in L_{1}(𝕋) or C(𝕋) is a natural example of a Markuschevich basis which is not a (Schauder) basis in any order (but this last statement is not easy to verify). Markuschevich bases do not always exist in the nonseparable setting, but are a useful tool for investigating nonseparable spaces in situations where they exist (see [44]). It is easy to see that any separable space X admits a Markuschevich basis. One starts with linearly independent sequences {y_{n}}_{n=1}^{∞} in X and {y_{n}^{∗}}_{n=1}^{∞} in X^{∗} with {y_{n}}_{n=1}^{∞} fundamental and {y_{n}^{∗}}_{n=1}^{∞} total and applies a Gram-Schmidt type procedure to biorthogonalize the sequences (alternating between working in X and X^{∗}) to produce a Markuschevich basis {x_{n}, x_{n}^{∗}}_{n=1}^{∞} so that span {x_{n}}_{n=1}^{∞} = span {y_{n}}_{n=1}^{∞} and span {x_{n}^{∗}}_{n=1}^{∞} = span {y_{n}^{∗}}_{n=1}^{∞} (see [14, 1.f.3]). A deeper fact ([14, 1.f.4]) is that a separable space contains a Markuschevich basis {x_{n}, x_{n}^{∗}}_{n=1}^{∞} for which sup_{n} ∥x_{n}∥∥x_{n}^{∗}∥ ≤ 20, and it is known that twenty can be replaced by any number larger than one. We shall see in section 8 that a finite dimensional space has a basis {x_{n}, x_{n}^{∗}}_{n=1}^{N} for which ∥x_{n}∥∥x_{n}^{∗}∥ = 1 for each n, but it is an open problem whether every separable space has a Markuschevich basis with this property. One separable theorem in which Markuschevich bases are used but do not appear in the statement is that every separable space X has a subspace Y such that both Y and X/Y have an FDD ([14, 1.g.2]). Incidentally, it is open whether the conclusion can be improved to “both Y and X/Y have a basis”.

4 Classical spaces

In this section we present some basic facts concerning the structure of the classical spaces C(K) and L_{p}(μ), their classification, and the relations among the spaces. Most books on real analysis, such as [18], contain some elementary results about the structure of these spaces. Beyond the most basic material,

[21] is accessible to students and has the additional advantage that some of the relations between these spaces and other spaces encountered in analysis are touched on. The more narrowly focused book [7] is also directed at students, contains several structural results omitted from [21], and the author’s off-the-wall style makes the book fun to read.

A normalized sequence {x_{n}}_{n=1}^{∞} of disjointly supported functions in L_{p}(μ), 1 ≤ p < ∞, is 1-equivalent (see section 3) to the unit vector basis for ℓ_{p} and the closed span of {x_{n}}_{n=1}^{∞} is the range of a norm one projection. The first statement is clear and the projection is defined by x ↦ ∑_{n} (∫ x|x_{n}|^{p−1}sign x_{n} dμ) x_{n}. In particular, every block basis of the unit vector basis for ℓ_{p}, 1 ≤ p < ∞, spans a space isometric to ℓ_{p} onto which there is a norm one projection. This last statement is also true if ℓ_{p} is replaced by c_{0}. Hence by the principle of small perturbations discussed in section 3, if Y is an infinite dimensional subspace of X, X = ℓ_{p} for 1 ≤ p < ∞ or X = c_{0}, then for any ϵ > 0 there is a subspace Z of Y with d(Z, X) < 1 + ϵ and so that there is a projection of norm less than 1 + ϵ from X onto Z.

Although general subspaces of X, X = ℓ_{p} for 1 ≤ p ≠ 2 < ∞ or X = c_{0} can be bad in that they can fail the approximation property and there are many of them, in a sense that is made precise in [24], the isomorphism theory of complemented subspaces of X is simple: An infinite dimensional complemented subspace Y of X, X = ℓ_{p} for 1 ≤ p < ∞ or c_{0}, is isomorphic to X. To see this, write X ≈ Y ⊕ V for some V. As mentioned in the previous paragraph, also Y ≈ X ⊕ W for some W. Hence (we write the formula for ℓ_{p}; for c_{0} the proof is identical but the notation changes)

\begin{aligned}ℓ_{p} & ≈ (ℓ_{p} ⊕ ℓ_{p} ⊕ ⋯)_{p} ≈ (Y ⊕ Y ⊕ ⋯)_{p} ⊕ (V ⊕ V ⊕ ⋯)_{p} \\ & ≈ Y ⊕ (Y ⊕ Y ⊕ ⋯)_{p} ⊕ (V ⊕ V ⊕ ⋯)_{p} ≈ Y ⊕ ℓ_{p} ≈ W ⊕ ℓ_{p} ⊕ ℓ_{p} ≈ Y. \end{aligned}

This method of proof, which is useful in many contexts, is called the decomposition method.

It is also true that every infinite dimensional complemented subspace of ℓ_{∞} is isomorphic to ℓ_{∞} but the proof requires an additional argument ([14, 2.a.7]).

Let 1 ≤ p < r < ∞. In section 3 we saw that every operator from a subspace of ℓ_{r} into ℓ_{p} is compact. There are noncompact operators from ℓ_{p} into ℓ_{r}, such as the formal identity operator I_{p,r}. When there exist “nonsmall” operators from one space to another (here “nonsmall” means “noncompact”, but elsewhere it can have another meaning), it is helpful to have one “nonsmall” operator which is typical of the genre. In fact, I_{p,r} is typical of the noncompact operators from ℓ_{p} into ℓ_{r} in a sense that can be made precise:

Say that an operator T : X → Y factors through an operator S : W → Z provided there exist operators U : X → W and V : Z → Y so that the diagram

\begin{matrix}W & ⟶^{S} & Z \\ U ↑ & & ↓ V \\ X & ⟶^{T} & Y \end{matrix}

commutes; that is, T = V SU. Now we can make precise the statement about I_{p,r}: If T is a noncompact operator from a subspace of ℓ_{p} into ℓ_{r}, 1 ≤ p < r < ∞, then I_{p,r} factors through T. Indeed, this is an immediate consequence of the comments in section 3 on noncompact operators from a subspace of a space with a basis and the remark in this section that a block basic sequence of the unit vector basis of ℓ_{p} spans a complemented copy of ℓ_{p} in ℓ_{p}.

It is easily seen that the only separable L_{∞}(μ) spaces are those for which the measure μ is purely atomic and has finitely many, say n, atoms, in which case L_{∞}(μ) is isometric to ℓ_{∞}^{n} since it is obvious that the indicator functions of n distinct atoms are, in L_{∞}(μ), 1-equivalent to the unit vector basis for ℓ_{∞}^{n}.

The isometric classification of the separable L_{p}(μ)-spaces, 1 ≤ p < ∞, is also simple. First, the measure μ can be assumed to be σ-finite. Secondly, any L_{p}(μ), 1 ≤ p ≤ ∞ with μ σ-finite, is isometric to a space of the same form with μ replaced by a probability measure. Indeed, take g positive a.e. [μ] with integral one and define a measure ν by dν = gdμ. Then the mapping f ↦ f · g^{−1/p} defines an isometry from L_{p}(μ) onto L_{p}(ν). (For p = ∞ the isometry is just the formal identity).

Next, if L_{p}(μ) is separable with μ a purely nonatomic probability measure, then the isomorphism theorem for separable measure algebras ([18, p. 399]) yields that L_{p}(μ) is isometric to L_{p}(0, 1). On the other hand, if the measure μ is purely atomic with atoms {A_{γ}}_{γ∈Γ}, then the function which maps for each γ the unit vector e_{γ} to μ(A_{γ})^{−1/p} times the indicator function of A_{γ} extends to an isometry from ℓ_{p}(Γ) onto L_{p}(μ). Now if x and y are in L_{p}(μ), 1 ≤ p ≠ 2 < ∞, then x and y are disjointly supported if and only if ∥x ± y∥_{p}^{p} = ∥x∥_{p}^{p} + ∥y∥_{p}^{p} ([18, p. 416]). This implies that an isometry from one L_{p} space onto another preserves disjointness and therefore also atoms. All these remarks combine to show that ℓ_{p}^{n}, ℓ_{p}, L_{p}(0, 1), ℓ_{p}⊕_{p}L_{p}(0, 1), ℓ_{p}^{n}⊕_{p}L_{p}(0, 1), n = 1, 2, … is a complete listing, up to isometry, of the separable L_{p}(μ) spaces when 1 ≤ p ≠ 2 < ∞, and these are all mutually nonisometric. Of course, in the Hilbertian case p = 2, ℓ_{2}^{n}, n = 1, 2, …; ℓ_{2} is the appropriate listing.

The decomposition method then yields that ℓ_{p}^{n}, ℓ_{p}, L_{p}(0, 1) is a complete listing, up to isomorphism, of the separable L_{p} spaces. Later in this section it is noted that ℓ_{p} is not isomorphic to L_{p}(0, 1) when 1 ≤ p ≠ 2 < ∞, so this is a listing of nonisomorphic spaces for these values of p.

In order to study the structure of subspaces of L_{p}(0, 1) as well as to investigate many other questions about L_{p} for one fixed value of p, it is convenient to consider the scale of L_{p} spaces as p varies. One reason for this is that (as we shall see) there are spaces of functions X on (0, 1) on which the norm ∥ · ∥_{p_{1}} is equivalent to ∥·∥_{p_{2}} with p_{1} < p_{2}. Note that when this occurs, since ∥·∥_{p} ≤ ∥·∥_{r} when p ≤ r, all the norms ∥ · ∥_{p} are equivalent on X for p_{1} ≤ p ≤ p_{2}. In fact, the extrapolation principle says that then all the norms ∥ · ∥_{p} are equivalent on X for p ≤ p_{2}. Indeed, suppose C is a constant so that ∥x∥_{p_{2}} ≤ C∥x∥_{p_{1}} for x ∈ X, 0 < p < p_{1}, and 0 < λ < 1 is defined by the formula p_{1} = λp+(1−λ)p_{2}. Then, by Hölder’s inequality, for x ∈ X we have

∥x∥_{p_{1}} ≤ ∥x∥_{p}^{λ}∥x∥_{p_{2}}^{(1−λ)} ≤ C^{1−λ}∥x∥_{p_{1}}^{1−λ}∥x∥_{p}^{λ}

and hence C^{1−1/λ}∥x∥_{p_{1}} ≤ ∥x∥_{p} ≤ ∥x∥_{p_{1}}.

The preceding yields particularly good information in case 2 < p and X is a closed subspace of L_{p}(0, 1) which is closed in L_{r}(0, 1) for some r ≠ p. By the open mapping theorem, the ∥ · ∥_{p} and ∥ · ∥_{r} norms are equivalent on X, and hence so is the ∥ · ∥_{2} norm. This means that X must be isomorphic to a Hilbert space. Moreover, the orthogonal projection P from L_{2}(0, 1) onto X induces a bounded linear projection P̃ from L_{p}(0, 1) onto X. The extrapolation principle then says that ∥ · ∥_{p^{∗}}, 1/p + 1/p^{∗} = 1, and ∥ · ∥_{2} are equivalent on X. The orthogonal projection onto X extends to the bounded projection P̃^{∗} from L_{p^{∗}}(0, 1) onto X.

Suppose that, on the other hand, X is a closed subspace of L_{p}(0, 1), 1 ≤ p < ∞, and for some (or, equivalently, every) 0 < r < p the ∥ · ∥_{p} and ∥ · ∥_{r} norms are not equivalent on X. Hölder’s inequality then yields that for any M < ∞ the infimum of ∥x1_{[|x|≤M]}∥_{p} as x ranges over the the unit sphere of X is zero. This means that X contains unit vectors which are an arbitrarily small perturbation of vectors whose supports have arbitrarily small measure, which in turn yields that X contains a sequence of unit vectors which is an arbitrarily small perturbation of a sequence of disjointly supported vectors in L_{p}(0, 1) and hence, by the principle of small perturbations, that X contains a subspace which is isomorphic to ℓ_{p} and complemented in L_{p}(0, 1).

These observations yield a dichotomy principle for subspaces of L_{p}(0, 1), 2 < p < ∞ [21, III.A.4]. Let X be a closed subspace of L_{p}(0, 1), 2 < p < ∞. Then either X is isomorphic to a Hilbert space and complemented in L_{p}(0, 1) or, for each C > 1, X contains a subspace C-isomorphic to ℓ_{p} and C-complemented in L_{p}(0, 1). This is stated for L_{p}(0, 1), but is valid for general L_{p}(μ) by the isometric classification of L_{p} spaces discussed earlier. Notice also that the comments in section 3 about ℓ_{p} now yield that if ℓ_{r} embeds isomorphically into L_{p}(μ), 0 < r < ∞ and 2 < p < ∞, then either r = p or r = 2.

The natural examples of infinite dimensional function spaces which are closed in L_{p} for more than one value of p come from probability theory. Suppose that {g_{n}}_{n=1}^{∞} is a sequence of independent standard Gaussian random variables, discussed in section 2. From the form of the distribution of g it is evident that ∥g∥_{p} is finite for all finite p and {g_{n}}_{n=1}^{∞} is an orthonormal sequence in L_{2}(0, 1). A characteristic function argument yields that if ∑_{k=1}^{n} |α_{k}|^{2} = 1, then ∑_{k=1}^{n} α_{k}g_{k} is again a standard Gaussian and hence has the same norm in L_{p}(0, 1) as g. This means that for all 0 < p < ∞, the mapping e_{n} ↦ ∥g∥_{p}^{−1}g_{n}, n = 1, 2, …, extends to an isometry from ℓ_{2} onto a subspace X of L_{p}(0, 1) where X does not even depend on p, and the orthogonal projection onto X defines a bounded projection from L_{p}(0, 1) onto X if 1 < p < ∞ (in section 10 we explain why no isomorph of ℓ_{2} in L_{1}(0, 1) or L_{∞}(0, 1) can be complemented). In particular, the dichotomy principle for subspaces of L_{p}, 2 < p < ∞, is not a monochotomy principle. This also implies, in view of the structure theory of ℓ_{p} discussed earlier, that ℓ_{p} and L_{p}(0, 1) are not isomorphic for 0 < p ≠ 2 < ∞.

A second natural example from probability theory of a subspace of L_{p}, 0 < p < ∞, which is isomorphic (but not isometric for p ≠ 2) to ℓ_{2} comes from considering a sequence {ε_{n}}_{n=1}^{∞} of independent random variables each taking on each of the values 1 and −1 with probability 1/2. Such a sequence {ε_{n}}_{n=1}^{∞} is called a Rademacher sequence. Classically the Rademacher functions {r_{n}}_{n=1}^{∞} are the concrete realization of such a sequence on [0, 1] defined by r_{n} := ∑_{k=0}^{2^{n}−1} h_{2^{n}+k}, where {h_{n}}_{n=1}^{∞} is the Haar system. Khintchine’s inequality says that a Rademacher sequence is equivalent, in the L_{p} norm, 0 < p < ∞, to the unit vector basis of ℓ_{2}; for future reference we write the inequality explicitly with the best constants labeled A_{p} and B_{p}:

A_{p} (∑ |α_{n}|^{2})^{1/2} ≤ (𝔼| ∑ α_{n}ε_{n}|^{p})^{1/p} ≤ B_{p} (∑ |α_{n}|^{2})^{1/2} . (1)

Since a Rademacher sequence is orthonormal, (∑ |α_{n}|^{2})^{1/2} = ∥ ∑ α_{n}ε_{n}∥_{2} ≡ (𝔼| ∑ α_{n}ε_{n}|^{2})^{1/2}, it follows that A_{p} = 1 for p ≥ 2 and B_{p} = 1 for p ≤ 2, and the exact values of A_{p} and B_{p} are known. The most elementary proof of Khintchine’s inequality proceeds by checking the right inequality in (1) for p an even integer. This clearly gives the result for all 2 ≤ p < ∞ and one then obtains the result for 0 < p < 2 from the extrapolation principle. The less computational modern proof of Khintchine’s inequality gives a vector valued version of Khintchine’s inequality called the Kahane-Khintchine inequality. This will be done in section 8.

The existence of an r-stable variable g (see section 2) for 0 < r < 2 shows that the subspace structure of L_{p}(0, 1), 0 < p < 2, is much more complicated than that of L_{p}(0, 1), 2 < p < ∞. Indeed, just as in the Gaussian case, there exists for 0 < r < 2 a sequence {g_{n}}_{n=1}^{∞} of symmetric r-stable random variables defined on (0, 1). If 0 < p < r, then these random variables are in L_{p}(0, 1). Now if ∑_{k=1}^{n} |α_{k}|^{r} = 1, then ∑_{k=1}^{n} α_{k}g_{k} is again symmetric r-stable and hence has the same norm in L_{p}(0, 1) as g. This means that for all 0 < p < r, the mapping e_{n} ↦ ∥g∥_{p}^{−1}g_{n}, n = 1, 2, …, extends to an isometry from ℓ_{r} into L_{p}(0, 1). In section 9 we explain how to derive from this the fact that for 1 ≤ p < r < 2, L_{r}(0, 1) embeds isometrically into L_{p}(0, 1). It turns out that this covers all the cases in which ℓ_{r} embeds isomorphically into L_{p}(μ) with p and r finite. The remaining cases are discussed in section 8.

In the reflexive range 1 < p < ∞, information about subspaces of L_{p^{∗}}(μ), 1/p + 1/p^{∗} = 1, given above gives information on quotients of L_{p}(μ). It turns out that the case p = 1 is quite different: Every separable Banach space X is isometric to a quotient of ℓ_{1}. Indeed, if {x_{n}}_{n=1}^{∞} is dense in the unit ball of X, the linear extension of the map e_{n} ↦ x_{n} (where {e_{n}}_{n=1}^{∞} is the unit vector basis for ℓ_{1}) maps the unit ball of ℓ_{1} onto a dense subset of the unit ball of X and hence extends to a quotient mapping from ℓ_{1} onto X. Similarly, if the Banach space X has density character κ, then X is isometric to a quotient of ℓ_{1}(Γ) when Γ has cardinality κ.

Another useful and interesting property of ℓ_{1} (which also holds for ℓ_{1}(Γ) for any set Γ) is the lifting property: If T is an operator from a Banach space X onto ℓ_{1} then there is a lifting S of T; that is, an operator S : ℓ_{1} → X for which T S = I_{ℓ_{1}}. Indeed, by the open mapping theorem there are x_{n} in X with T x_{n} = e_{n} and λ := sup ∥x_{n}∥ < ∞. The mapping e_{n} ↦ x_{n} then extends to an operator S : ℓ_{1} → X with ∥S∥ = λ satisfying T S = I_{ℓ_{1}}.

As noted in section 3, ℓ_{1} has the Schur property; that is, weakly convergent sequences in ℓ_{1} are norm convergent. An immediate consequence is that every weakly compact subset of ℓ_{1} is norm compact. The weakly compact subsets of L_{1}(0, 1) are more complicated since L_{1}(0, 1) contains infinite dimensional reflexive subspaces. There is however a nice characterization of subsets of L_{1}(μ) which have weakly compact closure when μ is a finite measure. First, if X is any Banach space and W is a subset of X such that for each ϵ > 0 there exists a weakly compact set S so that W ⊂ S + ϵB_{X}, then W has weakly compact closure (use the fact that a bounded subset of X is weakly compact if its weak^{∗} closure in X^{∗∗} is a subset of X). Next, given W ⊂ L_{1}(μ) with μ a finite measure, set for k ∈ ℕ, a_{k} = a_{k}(W) := sup{∥x1_{|x|≥k}∥_{1} : x ∈ W }. Clearly {a_{k}}_{k=1}^{∞} decreases to some a = a(W) ≥ 0. The set W is called uniformly integrable if a = 0. If a = 0 then W has weakly compact closure because W is a subset of kB_{L_{∞}(μ)} + a_{k}B_{L_{1}(μ)} and B_{L_{∞}(μ)} is weakly compact in L_{1}(μ).

If a(W) > 0, the set W does not have weakly compact closure and in fact even contains a sequence equivalent to the unit vector basis of ℓ_{1}. To see this, define two further numerical parameters for a subset W of L_{1}(μ); namely, b(W) := sup lim_{n→∞}{∥x_{n}1_{A_{n}}∥_{1}}, where the supremum is over all sequences {x_{n}}_{n=1}^{∞} in W and {A_{n}}_{n=1}^{∞} of measurable sets with μA_{n} → 0; and c(W), defined the same way as b(W) except that the supremum is over all sequences of disjoint measurable sets. It is an elementary exercise in measure theory to verify that a(W) = b(W) = c(W). Now if c(W) > 0, take a sequence {x_{n}}_{n=1}^{∞} in W and a sequence {A_{n}}_{n=1}^{∞} of disjoint measurable sets so that 0 < ∥x_{n}1_{A_{n}}∥_{1} → c(W). Clearly c({x_{n}1Ã_{n}}) = 0, so {x_{n}1Ã_{n}} has weakly compact closure. The sequence {x_{n}1_{A_{n}}}_{n=1}^{∞} is equivalent to the unit vector basis of ℓ_{1} since the A_{n}’s are disjoint and ∥x_{n}1_{A_{n}}∥ is bounded away from zero. This implies that a subsequence of {x_{n}}_{n=1}^{∞} is also equivalent to the unit vector basis of ℓ_{1}. (Remark: If {y_{n}}_{n=1}^{∞} is bounded in any L_{1}(μ) space and there is a sequence {B_{n}}_{n=1}^{∞} of disjoint measurable sets and b > 0 so that for all n, ∑_{k≠n} ∥y_{n}1_{B_{k}}∥_{1} ≤ b/2 < b ≤ ∥y_{n}1_{B_{n}}∥_{1}, then {y_{n}}_{n=1}^{∞} is equivalent to the unit vector basis of ℓ_{1}. Moreover, the closed span Y of {y_{n}}_{n=1}^{∞} is complemented in L_{1}(μ). Indeed, set B := ∪_{n=1}^{∞} B_{n}. The sequence {y_{n}1_{B}}_{n=1}^{∞} is a small perturbation of the disjoint sequence {y_{n}1_{B_{n}}}_{n=1}^{∞} and hence there is a projection P from L_{1}(μ) onto the closed span Z of {y_{n}1_{B}}_{n=1}^{∞}. The mapping y_{n}1_{B} ↦ y_{n} extends to an operator T from Z into the closed span of {y_{n}}_{n=1}^{∞} since {y_{n}1_{B}}_{n=1}^{∞} is equivalent to the unit vector basis of ℓ_{1}. Then f ↦ T P (f1_{B}) defines a projection onto Y.) To apply this remark one uses a combinatorial argument to verify that there ae n_{1} < n_{2} < … so that y_{k} := x_{n_{k}}, B_{k} := A_{n_{k}} satisfy the above conditions for any 0 < b < c(W).

It follows from the observations made in the beginning of this section that any separable subspace of any L_{1}(μ) space embeds isometrically into L_{1}(0, 1). Consequently, a bounded subset of L_{1}(μ) either has weakly compact closure or contains a sequence equivalent to the unit vector basis of ℓ_{1}. From this dichotomy principle and the fact that a weakly compact subset of a Banach space is weakly sequentially compact it follows that L_{1}(μ) is weakly sequentially complete. It also follows that a nonreflexive subspace of L_{1}(μ) contains a subspace which is isomorphic to ℓ_{1} and is complemented in L_{1}(μ).

There is a dichotomy principle, called Rosenthal’s ℓ_{1} theorem, for general Banach spaces which generalizes the one above for L_{1}: A bounded sequence in a Banach space either has a weakly Cauchy subsequence or a subsequence which is equivalent to the unit vector basis of ℓ_{1}. See [30] or [14, 2.5.e] or [7, Ch. XI].

A space Z is called λ-injective provided that for all Banach spaces X, every operator T from every subspace of X into Z has an extension to an operator T̃ from X into Z for which ∥T̃∥ ≤ λ∥T ∥. The spaces ℓ_{∞}(Γ) are 1-injective. This is proved by composing an operator T into Z with each of the evaluation functionals at points of Γ and applying the usual Hahn-Banach theorem. More generally, L_{∞}(μ) is 1-injective if μ is σ-finite. That is because L_{∞}(μ) is a complete lattice (see section 5) in this case and the usual proof of the scalar Hahn-Banach theorem carries over.

It is obvious that ℓ_{∞} embeds isometrically into L_{∞}(0, 1), and L_{∞}(0, 1) embeds isometrically into ℓ_{∞} because its predual, L_{1}(0, 1), is a quotient of ℓ_{1}, the predual of ℓ_{∞}. Since both ℓ_{∞} and L_{∞}(0, 1) are 1-injective, each space is isomorphic to a complemented subspace of the other. A simplified version of the decomposition method then shows that ℓ_{∞} and L_{∞}(0, 1) are isomorphic. Surprisingly, there is no known operator which exhibits explicitly the isomorphism between these two concrete spaces.

The space c_{0} has a weaker injectivity property: it is separably 2-injective, meaning that for all Banach spaces X and every operator T from a subspace Y of X with X/Y separable, there is an extension of T to an operator T̃ from X into c_{0} for which ∥T̃∥ ≤ 2∥T ∥. Indeed, since ℓ_{∞} is 1-injective, T extends to an operator T̃ from X into ℓ_{∞} having the same norm as T and the range of T̃ is separable because X/Y is separable. Thus it is enough to check that if Z is a separable space which contains c_{0}, then there is a projection of norm at most two from Z onto c_{0}. To see this, first let z_{n}^{∗} be an extension of the nth unit vector in ℓ_{1} = c_{0}^{∗} to a norm one element of Z^{∗}. Let d(·, ·) be a translation invariant metric on Z^{∗} which induces the weak^{∗} topology on the unit ball of Z^{∗}; for example, if {x_{n}}_{n=1}^{∞} is dense in the unit sphere of Z the metric can be defined by d(x^{∗}, y^{∗}) = ∑_{n=1}^{∞} 2^{−n}|(x^{∗} − y^{∗})(x_{n})|. Since every weak^{∗} limit point of {z_{n}^{∗}}_{n=1}^{∞} belongs to the unit ball B of the annihilator of c_{0} in Z^{∗}, it follows that d(z_{n}^{∗}, B) → 0. Thus we can choose w_{n}^{∗} in B so that d(z_{n}^{∗}, w_{n}^{∗}) → 0, which means that z_{n}^{∗} − w_{n}^{∗} → 0 weak^{∗}. The formula P z := {(z_{n}^{∗} − w_{n}^{∗})(z)}_{n=1}^{∞} defines a projection of norm at most two from Z onto c_{0}.

That the space c_{0} is not separably λ-injective for any λ < 2 can be seen by considering P 1 for any projection P from c onto c_{0}. More interesting is that the separability assumption is needed: There is no projection from ℓ_{∞} onto c_{0}. Indeed, otherwise we would have ℓ_{∞} ∼ c_{0} ⊕ ℓ_{∞}/c_{0}. But ℓ_{∞}/c_{0} is not isomorphic to a subspace of ℓ_{∞}. This can be seen by proving that c_{0}(ℝ) embeds into ℓ_{∞}/c_{0} but not into ℓ_{∞}. c_{0}(ℝ) admits no countable separating family of linear functionals, so it does not embed into ℓ_{∞}. To see that c_{0}(ℝ) embeds into ℓ_{∞}/c_{0}, take a family {A_{r}}_{r∈ℝ} of infinite subsets of ℕ so that the intersection of any two is finite (replace ℕ by the rationals and for each r ∈ ℝ consider a sequence of rationals which converges to r). Let x_{r} be the image in the quotient space ℓ_{∞}/c_{0} of the indicator function of A_{r}. It is easy to check that {x_{r}}_{r∈ℝ} is 1-equivalent to the unit vector basis of c_{0}(ℝ).

The space c_{0} is the only (up to isomorphism) separable space which is separably injective [43]. General C(K) spaces do have another useful into extension property: A compact operator T from a subspace Y of a Banach space X into C(K) has an extension T̃ to a compact operator from X into C(K). Indeed, since C(K) spaces have the BAP, T can be approximated in the operator norm by operators of finite rank and hence we can write T = ∑_{n=0}^{∞} T_{n} with each T_{n} of finite rank and ∥T_{n}∥ < 2^{−n} for n ≥ 1. Therefore it is enough to observe that if S : X → C(K) has finite rank, then S extends to a finite rank operator S̃ : X → C(K) with ∥S̃∥ ≤ (1+ϵ)∥S∥ (where ϵ > 0 is arbitrary). But by using partitions of unity and the principle of small perturbations one checks that there is a subspace E of C(K) so that SX ⊂ E and d(E, ℓ_{∞}^{n}) < 1 + ϵ, where n = dim E < ∞. Then E is (1 + ϵ)-injective and hence the desired extension of S exists. The same argument works when C(K) is replaced by any L_{∞} space (defined in section 9).

Every L_{∞}(μ) is isometric to C(K) for some compact Hausdorff space K. This follows from Gelfand theory, since L_{∞}(μ) is a commutative B^{∗} algebra with unit (see [19, Ch. 11]). Alternatively, it follows from lattice characterizations of C(K) spaces (see section 5).

The C(K) spaces play a special rôle in Banach space theory because they are a universal class: Every Banach space X is isometric to a subspace of some C(K) space. This can be seen by embedding X into ℓ_{∞}(Γ) for Γ appropriately large and applying the comment in the previous paragraph. Alternatively, X isometrically embeds via evaluation into C(K) with K the unit ball of X^{∗} with the weak^{∗} topology. This approach is preferable because the unit ball of X^{∗} is weak^{∗} metrizable when X is separable. In fact, every separable Banach space X embeds isometrically into C[0, 1]. First, C(Δ) embeds isometrically into C[0, 1], where Δ is the usual “middle thirds” Cantor set in [0, 1], by extending a continuous function on Δ affinely on the component intervals of the complement of Δ in [0, 1]. Next, using that Δ is homeomorphic to {0, 1}^{ℕ}, it can be shown that every compact metric space K is the continuous image of Δ. Build a tree structure K(i_{1}, i_{2}, …, i_{n}) : {i_{1}, i_{2}, …, i_{n}} ∈ {0, 1}^{n}; n = 1, 2, …} of nonempty closed subsets of K so that each K(i_{1}, i_{2}, …, i_{n}) is the union of K(i_{1}, i_{2}, …, i_{n}, 0) and K(i_{1}, i_{2}, …, i_{n}, 1), K = K_{0} ∪K_{1}, and the maximum diameter of K(i_{1}, i_{2}, …, i_{n}) as {i_{1}, i_{2}, …, i_{n}} varies over {0, 1}^{n} tends to zero as n → ∞. This is not hard to do using compactness of K. The map defined by {i_{j}}_{j=1}^{∞} ↦ K(i_{1}) ∩ K(i_{1}, i_{2}) ∩ K(i_{1}, i_{2}, i_{3}) ∩ … is then a continuous surjection from Δ = {0, 1}^{ℕ} onto K. Having gotten a continuous mapping g from Δ onto K, one embeds C(K) into C(Δ) by the formula T x = x ∘ g, x ∈ C(K).

The isometric classification of C(K) spaces is easy and can be found in beginning texts (such as [12, Th. 231]): The space C(K) is isometric to C(H) if and only if K is homeomorphic to H. This follows from the identification of the extreme points of the unit ball of C(K)^{∗} as the evaluation functionals at points of K (as well as multiples of these functionals by scalars of magnitude one). The much harder problem of the isomorphic classification of C(K) spaces has been accomplished in the separable case (that is, for compact metric spaces): If K is an uncountable compact metric space then C(K) is isomorphic to C(0, 1). If K is a countable compact metric space then K is homeomorphic to the space [1, α] of all ordinals up to the ordinal α for some countable ordinal α in the order topology. C(1, α) is isomorphic to C(1, β) when α < β if and only if β < α^{ω}. See [40] for further discussion of the separable case. It seems a hopeless task to get an isomorphic classification of general C(K) spaces, but some information is contained in [44].

We have already mentioned inexplicitly that the dual of an L_{1}(μ) space is isometric to a C(K) space. Similarly, the dual of a C(K) is isometric to L_{1}(μ) for some measure μ. The usual representation (see [18]) of the dual of C(K) is the space M(K) of finite signed measures on the sigma algebra of Baire subsets of K (that is, the sigma algebra generated by the closed G_{δ} subsets of K). It is a routine exercise using the Radon-Nikodým theorem to verify that if {μ_{γ}}_{γ∈Γ} is a maximal family of mutually singular Baire probability measures on K, then M(K) is isometric to (∑_{γ∈Γ} L_{1}(μ_{γ}))_{1}.

Once the duals of C(K) and of L_{1}(μ) are classified, it is natural to ask what are the preduals of C(K) and L_{1}(μ)? It turns out that every isometric predual of a space C(K) is isometric to L_{1}(μ) for some measure μ and that all preduals of C(K) are mutually isometric. On the other hand, the isomorphic preduals of, for example, ℓ_{∞}, form a rich class of spaces which are still not well understood. Preduals of L_{1}(μ) spaces are even less well understood. Here we mention only that the space ℓ_{1} has uncountably many mutually nonisomorphic preduals among the C(K) spaces (C(K)^{∗} is isometric to ℓ_{1} if K is a countable compact space).

5 Banach lattices

Most of the Banach spaces that appear naturally in analysis carry structure in addition to their structure as Banach spaces. It turns out that this additional structure can affect properties which are defined purely in terms of Banach space concepts (such as linear operators and duality). An extra structure that classical spaces such as L_{p} and C(K) spaces possess is that of a Banach lattice, which is a Banach space over the reals that is equipped with a partial order ≤ for which x ∨ y and x ∧ y exist for all vectors x, y, and such that the positive cone is closed under addition and multiplication by nonnegative real numbers and the order is connected to the norm by the condition that |x| ≤ |y| ⇒ ∥x∥ ≤ ∥y∥, where the absolute value is defined by |x| = x ∨ (−x).

A linear mapping from a Banach lattice to a Banach lattice is positive if it carries positive vectors to positive vectors. This is equivalent to saying that the mapping is order preserving. It is easy to see that a positive linear mapping is continuous, so we call them positive operators. If a positive operator preserves the lattice operations, it is called a lattice homomorphism. It is clear that a one-to-one surjective positive operator T whose inverse is positive is a lattice isomorphism; that is, both T and T^{−1} are lattice homomorphisms. The dual of a Banach lattice X is again a Banach lattice under the standard ordering on X^{∗} in which the positive linear functionals form the positive cone. With this definition it is easy to see that the canonical mapping from X into X^{∗∗} is positive ([15, 1.a.2]).

While there are interesting functional analytical topics (such as positive operators and lattice homomorphisms) which are special to Banach lattices, we are mostly interested here in the Banach space properties of Banach lattices and concentrate on those lattice properties which affect the linear topological or geometry of the underlying Banach space. The basic reference for this aspect of Banach lattices is [15]. For a general introduction to Banach lattices which covers the basic theory as well as some fairly recent material see [1].

The simplest examples of Banach lattices are the spaces with a monotonely unconditional basis under the pointwise order on the coefficients, which as we have seen are better behaved, or at least more regularly behaved, than general Banach spaces. It turns out that much of the structure theory for these spaces carries over to Banach lattices.

It was already mentioned that the L_{p} and C(K) spaces are Banach lattices. Other examples are the Orlicz spaces and the Lorentz spaces. An Orlicz function is an even convex function on ℝ which is zero at zero and tends to infinity at infinity. If μ is a measure and M is an Orlicz function, the space L_{M}(μ) is the collection of all μ-measurable functions f for which there exists C > 0 so that ∫ M(f/C) dμ < ∞. Then ∥f∥_{M} is defined to be the infimum of those C > 0 for which ∫ M(f/C) dμ < 1. This is a norm on L_{M}(μ) which makes L_{M}(μ) into a Banach lattice. If μ is counting measure on ℕ, L_{M}(μ) is called an Orlicz sequence space and is denoted by ℓ_{M}. If 1 ≤ p < ∞ and W is a positive nonincreasing continuous function on (0, ∞) so that W (t) → ∞ as t → 0+, W (t) → 0 as t → ∞, ∫_{0}^{1} W (t) dt = 1, and ∫_{0}^{∞} W (t) dt = ∞, the Lorentz space L_{W,p}(μ) is the space of all μ measurable functions f for which ∥f∥_{W,p} := (∫_{0}^{∞} f^{∗}(t)^{p}W (t) dt)^{1/p}, where f^{∗} is the decreasing rearrangement of |f|. The space L_{W,p}(μ) is a Banach lattice under the norm ∥f∥_{W,p}. Lorentz sequence spaces are defined in an analogous fashion.

The Orlicz spaces L_{M}(μ) and the Lorentz spaces L_{W,p}(μ), like the spaces L_{p}(μ), are symmetric lattices which are ideals in the lattice of all μ-measurable functions. Here X is symmetric means that if x ∈ X and y is a μ-measurable function for which x^{∗} = y^{∗}, then y is in X and ∥y∥_{X} = ∥x∥_{X}. A subspace Y of a lattice X is an ideal provided x ∈ X, y ∈ Y, and |x| ≤ |y| ⇒ x ∈ Y. Symmetric lattice ideals play an important role in interpolation theory; see section (11) for a short introduction. (Some sources call a symmetric lattice a rearrangement invariant space; other references use “rearrangement invariant space” to mean “symmetric lattice ideal”, sometimes with extra conditions. That is why we avoid the term “rearrangement invariant” in this article.)

It is no accident that in the examples of Banach lattices given above the lattice ordering is just the natural pointwise a.e. ordering on a space of (equivalence classes of) real valued functions on a measure space (for C(K) the measure is counting measure on K and for spaces with unconditional basis the measure is counting measure on ℕ). We call such a Banach lattice a Banach lattice of functions or a Banach lattice of μ-measurable functions if it is important to specify the measure. There are representation theorems which say that there is essentially no loss of generality in considering only Banach lattices of functions. The most classical of these representation theorems, called the Kakutani representation theorem, gives abstract Banach lattice characterizations of L_{1}(μ) and C(K) spaces. A Banach lattice X is an abstract L_{p} space, 1 ≤ p < ∞, provided ∥x + y∥^{p} = ∥x∥^{p} + ∥y∥^{p} whenever x and y are disjoint; that is, |x| ∧ |y| = 0. The L_{p} version of the Kakutani representation theorem says that an abstract L_{p} space is lattice isometric to L_{p}(μ) for some measure μ (see [15, 1.b.2] or [1, Th 12.26]). Moreover, the measure μ can be chosen to be a finite measure if the abstract L_{p} space X has a weak order unit; that is, a vector u ≥ 0 so that u ∧ |x| = 0 only when x is the zero vector. A Banach lattice X is an abstract M space provided ∥x + y∥ = ∥x∥ ∨ ∥y∥ whenever x and y are disjoint. The C(K) representation theorem says that an abstract M space is lattice isometric to a sublattice of C(K) for some compact Hausdorff space K (see [15, 1.b.6]). Moreover, if the abstract M space has a strong order unit (that is, a vector u ≥ 0 such that the unit ball of X is the order interval [−u, u] := {x : −u ≤ x ≤ u}), then M is lattice isometric to C(K) itself via an isomorphism which maps u to ∥u∥1_{K}.

Suppose that the Banach lattice X admits a strictly positive functional; that is, a positive linear functional u^{∗} such that u^{∗}(|x|) > 0 for every nonzero vector x in X. Define an (inequivalent) norm ∥ · ∥_{u^{∗}} on X by ∥x∥_{u^{∗}} := u^{∗}(|x|). This is obviously an (incomplete) abstract L_{1} norm on X from which it follows that the completion X_{u^{∗}} of (X, ∥x∥_{u^{∗}}) is an abstract L_{1} space. Since the formal inclusion operator from X to X_{u^{∗}} is an injective lattice homomorphism, the Kakutani representation theorem yields that X can be thought of as a Banach lattice of functions. If X is separable, so is X_{u^{∗}}, and it follows from comments made in section 4 that X_{u^{∗}} is isometric and order equivalent to a sublattice of L_{1}(0, 1), so that X can be represented as a Banach lattice of Lebesgue measurable functions on the unit interval.

Not every Banach lattice admits a strictly positive functional (c_{0}(Γ) with Γ uncountable is a counterexample), but every separable Banach lattice X does. Indeed, take {x_{n}}_{n=1}^{∞} dense in the unit sphere of X and for each n pick a norm one functional x_{n}^{∗} which achieves its norm at x_{n}. It is easy to check that u^{∗} := ∑ 2^{−n}|x_{n}^{∗}| is a strictly positive functional. From this comment and the representation above it follows that any lattice inequality involving finitely many vectors which is true for lattices of functions must be true in a general Banach lattice. This eliminates the tedium of verifying “obvious” inequalities (for example, ∨{∑_{i=1}^{n} ϵ_{i}x_{i} : ϵ_{i} = ±1} = ∑_{i=1}^{n} |x_{i}|) directly from the axioms for a Banach lattice.

The representation of Banach lattices as lattices of functions suggests that abstract L_{1} spaces are particularly important lattices. The abstract M spaces also arise naturally in the general theory. If u is a positive vector in a Banach lattice X and X_{u} is the linear span of the order interval [−u, u] with [−u, u] taken as the unit ball, then X_{u} is easily seen to be an abstract M space with strong order unit u (X_{u} is complete because [−u, u] is closed in the Banach space X).

A Banach lattice is order complete or Dedekind complete if every nonempty subset which is bounded above has a least upper bound. A dual Banach lattice is order complete; in fact, any norm bounded upwarded directed net in a dual Banach lattice converges weak^{∗} to the least upper bound of the net.

Suppose that X is order complete and u ≥ 0 with ∥u∥ = 1. We look a bit more closely at the C(K) space which is lattice isometric to the abstract M space X_{u}. Given x ≥ 0, define the support of x by S(x) := sup_{n}(nx) ∧ u.

The supremum exists because X is order complete. The support S(x) is a component of u. (A vector 0 ≤ y ≤ u is called a component of u [or simply a component if u is understood] provided y is disjoint from u − y.) If T : X_{u} → C(K) is a lattice isometry with T u = 1_{K}, then T x is an indicator function if and only if x is a component. Of course, if A ⊂ K, 1_{A} is in C(K) if and only if A is clopen (i.e., both open and closed). So the components form a Boolean algebra (a fact which is also easy to verify directly) which is complete because X is order complete. Thus the clopen subsets of K are also a complete Boolean algebra. An important fact is that the clopen subsets of K form a base for the topology of K. One way to see this is to observe that if 0 ≤ x with x in X_{u} and t > 0 satisfies y := (x − tu) ∨ 0 ≠ 0, then 0 ≤ tS(y) ≤ x. The interpretation of this fact in C(K) is that for all 0 ≤ f in C(K) with f ≠ 0, the set [f > 0] contains a nonempty clopen subset. Since K is compact this implies that the clopen subsets of K form a base for the topology. Using this and the completeness of the Boolean algebra of clopen sets it is a simple exercise to prove that C(K) is itself an order complete Banach lattice. As was pointed out in section 4, this implies that C(K) is 1-injective.

From the discussion in the previous paragraph we can deduce: If E is a finite dimensional subspace of an order complete lattice X and ϵ > 0, then there is a finite dimensional sublattice F of X and an automorphism T of X so that E ⊂ T F and ∥I − T ∥ < ϵ. Given the subspace E, take positive vectors x_{1}, …, x_{n} in X whose span contains E and normalized so that u := max_{i} x_{i} has norm one. Then F ⊂ X_{u} and X_{u} is isometric to a C(K) space for which the clopen subsets of K form a base for the topology, which means that the span of indicator functions of clopen sets is dense in C(K). Thus fixing a basis y_{1}, …, y_{k} for E and δ > 0, we get a subspace F of X_{u} spanned by disjoint vectors and a vector u in F with ∥u∥ = 1 so that for each 1 ≤ i ≤ k, there is a vector x_{i} so that |x_{i} − z_{i}| ≤ δu (and hence ∥x_{i} − z_{i}∥ ≤ δ). Now apply the principle of small perturbations.

A Banach lattice is order continuous if every downward directed net whose greatest lower bound is zero converges in norm (or weakly; it is the same) to zero. This is equivalent to saying that every order bounded increasing sequence converges in norm (necessarily to the least upper bound of the sequence) (see [15, 1.a.8] or [1, Th.12.9] or the beginning of the argument below). It is also easy to check that an order continuous Banach lattice is order complete [15, 1.a.8]. A Banach lattice is not order continuous if and only if it contains a sequence of disjoint positive vectors which is equivalent to the unit vector basis for c_{0} and is bounded above. The “if” direction is clear. If X is not order continuous, one gets an upward directed net of positive vectors which is bounded above by, say, x, with ∥x∥ = 1. The net cannot converge in norm, so one gets 0 ≤ x_{1} ≤ x_{2} ≤ … ≤ x so that a := inf_{n} ∥x_{n+1} − x_{n}∥ > 0. Let y_{n} := x_{n+1} − x_{n} ≥ 0. So we have for every n:

y_{n} ≥ 0, ∥y_{n}∥ ≥ a, ∑_{k=1}^{n} y_{k} ≤ x. (2)

(Although we do not need it here because we want to “disjointify” the y_{n}’s, it is worth noticing that (2) implies that y_{n} → 0 weakly and hence {y_{n}}_{n=1}^{∞} has a basic subsequence, and that any basic subsequence of {y_{n}}_{n=1}^{∞} is equivalent to the unit vector basis for c_{0}.) Take y_{n}^{∗} in the unit sphere of X^{∗} so that y_{n}^{∗}(y_{n}) = ∥y_{n}∥. By replacing y_{n}^{∗} with |y_{n}^{∗}| if necessary, we may assume that y_{n}^{∗} ≥ 0. For each n the nonnegative sum ∑_{m} y_{n}^{∗}(y_{m}) is at most ∥x∥. It is an elementary exercise in combinatorial reasoning to deduce from this that for any ϵ > 0 there is a subsequence {z_{k}}_{k=1}^{∞} := {y_{n_{k}}}_{k=1}^{∞} of {y_{n}}_{n=1}^{∞} so that for each k > j, z_{k}^{∗}(z_{j}) < 2^{−j}ϵ^{2}, where z_{k}^{∗} := y_{n_{k}}^{∗}. Thus for each n,

∥ (z_{n+1} − ϵ^{−1} ∑_{k=1}^{n} z_{k}) ∨ 0∥ ≥ z_{n+1}^{∗} (z_{n+1} − ϵ^{−1} ∑_{k=1}^{n} z_{k}) ≥ a − ϵ.

This means that the z_{n}’s have big disjoint pieces. More precisely, let ϵ = a/4 and set v_{n+1} := (z_{n+1} − ϵ^{−1} ∑_{k=1}^{n} z_{k}) ∨ 0 and w_{n} := (v_{n} − ϵx) ∨ 0. Then the w_{n}’s are pairwise disjoint positive vectors smaller than x with norms bounded away from zero.

It is easy to see that {w_{n}}_{n=1}^{∞} must be equivalent to the unit vector basis of c_{0}. This completes the proof, but note that if X is order complete the mapping e_{n} → w_{n} from c_{0} into X extends to a mapping from the positive cone of ℓ_{∞} to X by defining {α_{n}δ_{n}}_{n=1}^{∞} ↦ sup_{n} α_{n}w_{n}; this mapping extends to an order isomorphism from ℓ_{∞} into X.

Other useful characterizations of order continuous Banach lattices are given by the following (see [15, 1.b.16] or [1, Th. 12.9]): X is order continuous if and only if every order interval is weakly compact if and only if X is an ideal in X^{∗∗}.

Since c is not order continuous but is isomorphic to the order continuous Banach lattice c_{0}, there is not a linear topological characterization of order continuity. While it is only a sufficient condition for order continuity of X that c_{0} not embed isomorphically into X, this condition is only a bit too strong as arguments similar to those given above show that if c_{0} embeds into X then it embeds as a sublattice (see [1, Th. 14.12]).

The functional representation theorems for order continuous Banach lattices are stronger and more useful than the representation already mentioned for Banach lattices which have a strictly positive functional. Here we just indicate what is going on and refer to [15] for details. However, the motivated reader is encouraged to work out the details for himself or herself (partly because the discussion in [15] wanders unnecessarily). First, it is not hard to show that an order continuous Banach lattice which has a weak order unit u also admits a strictly positive functional u^{∗} (see [15, 1.b.15] or [1, 12.14]) and thus can be thought of as a space of integrable functions on a finite measure space (Ω, μ) with u^{∗}(x) = ∫ x dμ for x in X. One can assume that u^{∗}(u) = 1 = ∥u∥ and ∥u^{∗}∥ ≤ 2. Since X is order continuous, one has for each x ≥ 0 in X that (nu) ∧ x converges to x as n → ∞, so that X_{u} is dense in X. By replacing the underlying σ-algebra with the smallest σ-algebra for which all the functions in X are measurable, it can be assumed that X is dense in L_{1}(μ). Then necessarily u > 0 a.e. By replacing the measure μ with u dμ and functions f in X by f/u, one can assume u ≡ 1. Now it is possible to recover the μmeasurable sets and the measure μ. From the density of X in L_{1}(μ) it follows that the indicator function of a measurable set is an element x in X_{u} which is a component of u. From this it is essentially obvious that the L_{∞}(μ) norm on X_{u} agrees with its abstract M space norm. If we represent X_{u} as a C(K) space, then the sets of positive μ measure are mapped onto the nonempty clopen subsets of K in an obvious way so that we can represent X and L_{1}(μ) as function spaces on K and transfer the measure μ to K. We then get that C(K) ⊂ X ⊂ L_{1}(μ) with both inclusions having dense range. Moreover, C(K) = L_{∞}(μ) and every μ-measurable set is equal μ-a.e. to a clopen subset of K. Finally, X is an ideal in L_{1}(μ) (use again order continuity and the fact that X contains all indicator functions).

One consequence of this representation is: Let X be an order continuous Banach lattice which has a weak order unit. A closed subspace Y of X either embeds into L_{1}(μ) for some measure μ or contains a normalized basic sequence which is equivalent to (even equal to a small perturbation of) a disjoint sequence. For a proof see [15, 1.c.8] or modify the proof of the dichotomy principle discussed in section 4. By applying L_{1} theory discussed in section 4 one gets that a subspace of an order continuous Banach lattice is reflexive if and only if it does not contain a subspace isomorphic to either ℓ_{1} or c_{0} and that a nonreflexive Banach lattice has a sublattice which is order isomorphic to ℓ_{1} or c_{0}. The representation can also be used to prove that a Banach lattice which does not contain a copy of c_{0} must be weakly sequentially complete ([15, 1.c.4]).

Although we have used here the representation theorem for abstract L_{1} spaces, it should be mentioned that since an abstract L_{1} space is necessarily order continuous (every normalized disjoint sequence is obviously 1-equivalent to the unit vector basis of ℓ_{1}), some of the ideas (particularly building components in X) can be used to prove Kakutani’s representation of an abstract L_{1} space which has a weak order unit as a space L_{1}(μ) for some probability measure μ.

Expressions such as (∑_{n=1}^{N} |x_{n}|^{p})^{1/p}, 1 ≤ p ≤ ∞, have meaning in a general Banach lattice. (When p = ∞ we follow the convention of interpreting (∑_{n=1}^{N} |x_{n}|^{p})^{1/p}to be ∨_{n=1}^{N}|x_{n}|.) Letting 1/p + 1/p^{∗} = 1, one can define

(∑_{n=1}^{N} |x_{n}|^{p})^{1/p} = ∨{∑_{n=1}^{N} α_{n}x_{n} : ∑_{n=1}^{N} α_{n}^{p^{∗}} = 1}, but one must justify that this supremum of an infinite set of vectors must exist in the (possibly lattice incomplete) Banach lattice. By taking the supremum over finite sets of scalars satisfying ∑_{n=1}^{N} α_{n}^{p^{∗}} = 1 one obtains an upward directed net which is bounded above by ∑_{n=1}^{N} |x_{n}|; it is enough to check that this net converges in X. This is not hard to do directly, but note that if X is a C(K) space the net obviously converges to (∑_{n=1}^{N} |x_{n}|^{p})^{1/p}. By working in the abstract M space X_{u} with u := ∑_{n=1}^{N} |x_{n}|, the C(K) space case gives the general case. We can then interpret scalar inequalities involving such expressions in general lattices (rather than just in lattices of functions). For example, Khintchine’s inequality, discussed in section 4, reads

A_{p} (∑_{n=1}^{N} |x_{n}|^{2})^{1/2} ≤ (𝔼| ∑_{n=1}^{N} ε_{n}x_{n}|^{p})^{1/p} ≤ B_{p} (∑_{n=1}^{N} |x_{n}|^{2})^{1/2} . (3)

Auxiliary lattices such as X(ℓ_{p}) can now be defined and the “obvious” dualities checked. X(ℓ_{2}^{2}) is especially noteworthy, since it can be regarded as the complexification of X by regarding ℓ_{2}^{2} as ℂ and defining multiplication by complex scalars in the obvious way (see [15, p. 43] for details).

Starting from the scalar identity |α|^{θ}|b|^{1−θ} = inf_{C>0} θC^{1/θ}|α|+(1−θ)C^{1/(θ−1)}|β|, one gets in a similar way an interpretation for the expression |x|^{θ}|y|^{1−θ} and that

∥|x|^{θ}|y|^{1−θ}∥ ≤ ∥x∥^{θ}∥y∥^{1−θ}. (4)

Also, if 1 ≤ p < q ≤ ∞, 1/r := θ/p + (1 − θ)/q with 0 < θ < 1 and α_{1}, …, α_{N} are positive scalars, then for any collection x_{1}, …, x_{N} of vectors in a Banach lattice we have that

(∑_{n=1}^{N} α_{n}|x_{n}|^{r})^{1/r} ≤ (∑_{n=1}^{N} α_{n}|x_{n}|^{p})^{θ/p} (∑_{n=1}^{N} α_{n}|x_{n}|^{q})^{(1−θ)/q} . (5)

Inequality (5) is obvious for lattices of functions, as is the following: If 1 ≤ p ≤ ∞, x_{1}, …, x_{N} are vectors in a Banach lattice X and x_{1}^{∗}, …, x_{N}^{∗} are in X^{∗}, then

∑_{n=1}^{N} x_{n}^{∗}(x_{n}) ≤ (∑_{n=1}^{N} |x_{n}^{∗}|^{p})^{1/p} (∑_{n=1}^{N} |x_{n}|^{p^{∗}})^{1/p^{∗}} . (6)

A more sophisticated version [15, 1.d.1] of the argument given above shows that for any continuous homogeneous (f(λx) = λf(x) if λ ≥ 0) function f on ℝ^{n} and x_{1}, …, x_{n}, the expression f(x_{1}, …, x_{n}) can be defined in such a way that any lattice inequality that is true in ℝ^{n} is true in X; that is, if g is another continuous homogeneous function on ℝ^{n} and f(a) ≤ g(a) for all a = (α_{1}, …, α_{n}) in ℝ^{n}, then f(x_{1}, …, x_{n}) ≤ g(x_{1}, …, x_{n}) for all x_{1}, …, x_{n} in X.

For 1 ≤ p ≤ ∞, a linear mapping T from a Banach space into a Banach lattice is called p-convex if there exists a constant M so that for all finite sets of vectors in the domain space the following inequality holds:

∥(∑_{n=1}^{N} |T x_{n}|^{p})^{1/p}∥ ≤ M (∑_{n=1}^{N} ∥x_{n}∥^{p})^{1/p} . (7)

The smallest such M is denoted by M^{(p)}(T). Clearly M^{(1)}(T) = ∥T ∥. Similarly, if for a linear mapping T from a Banach lattice into a Banach space the inequality

(∑_{n=1}^{N} ∥T x_{n}∥^{p})^{1/p} ≤ M ∥(∑_{n=1}^{N} |x_{n}|^{p})^{1/p}∥ (8)

always holds for some constant M, then T is called p-concave and the smallest such M is denoted by M_{(p)}(T). Clearly M_{(∞)}(T) = ∥T ∥. Evidently if T is pconvex [p-concave] then it is bounded and M^{(p)}(T) ≥ ∥T ∥ [M_{(p)}(T) ≥ ∥T ∥]. It is not hard to check the identities M^{(p)}(T^{∗}) = M_{(p^{∗})}(T) and M^{(p)}(T) = M_{(p^{∗})}(T^{∗}), where 1/p + 1/p^{∗} = 1. As functions of p, M^{(p)}(T) is nondecreasing and M_{(p)}(T) is nonincreasing (see [15, 1.d.5] or write down a proof for L_{p}(μ) and see that the Hölder type inequalities (4), (5) allow a translation to the lattice setting).

A Banach lattice X is called p-convex [p-concave] if the identity operator I_{X} on X is p-convex [p-concave] and we then define M^{(p)}(X) := M^{(p)}(I_{X}) and M_{(p)}(X) := M_{(p)}(I_{X}). These constants are called the p-convexity and pconcavity constants of X. So X is p-convex [p-concave] if and only if X^{∗} is p^{∗}-concave [p^{∗}-convex].

A p-convex and r-concave Banach lattice can be renormed with an equivalent lattice norm so that the p-convexity and r-concavity constants are both one [15, 1.d.8]. In particular, a lattice which is both p-convex and p-concave is lattice isomorphic to an abstract L_{p} space.

If T : X → Y is a positive operator between Banach lattices it is easy to check (see [15, 1.d.9]) the inequality

∥(∑_{n=1}^{N} |T x_{n}|^{p})^{1/p}∥ ≤ ∥T ∥ ∥(∑_{n=1}^{N} |x_{n}|^{p})^{1/p}∥ . (9)

From this it follows that M^{(p)}(T) ≤ ∥T ∥M^{(p)}(X) and M_{(p)}(T) ≤ ∥T ∥M_{(p)}(Y). There are characterizations [15, 1.d.10] of general operators which are p-convex or p-concave which involve the notion of p-summing operator, to be discussed in section 10. When p = 2, the inequality (9) is valid for general operators (except that the right hand side must be multiplied by a constant; see section 10 or [15, 1.f.14]).

The square function (∑_{n=1}^{N} |x_{n}|^{2})^{1/2} of vectors x_{1}, …, x_{N} in a p-concave, 1 ≤ p < ∞, lattice X is very useful for studying unconditional basic sequences and other things as well. The reason is that by taking norms in the lattice Khintchine inequality (3) we get

\begin{aligned}A_{1}∥(∑_{n=1}^{N} |x_{n}|^{2})^{1/2}∥ & ≤ ∥𝔼 | ∑_{n=1}^{N} ε_{n}x_{n}|∥ ≤ \\ 𝔼 ∥ ∑_{n=1}^{N} ε_{n}x_{n}∥ & ≤ (𝔼 ∥ ∑_{n=1}^{N} ε_{n}x_{n}∥^{p})^{1/p} . \end{aligned}

By p-concavity, this last quantity is dominated by

M_{(p)}(X)∥ (𝔼 | ∑_{n=1}^{N} ε_{n}x_{n}|^{p})^{1/p} ∥ ≤ M_{(p)}(X)B_{p}∥ (∑_{n=1}^{N} |x_{n}|^{2})^{1/2} ∥.

This gives the equivalence of the norm of the square function of x_{1}, …, x_{n} with a certain Rademacher average:

\begin{matrix}A_{1}∥ (∑_{n=1}^{N} |x_{n}|^{2})^{1/2} ∥ & ≤ 𝔼 ∥ ∑_{n=1}^{N} ε_{n}x_{n}∥ ≤ (𝔼 ∥ ∑_{n=1}^{N} ε_{n}x_{n}∥^{p})^{1/p} \\ & ≤ M_{(p)}(X)B_{p}∥ (∑_{n=1}^{N} |x_{n}|^{2})^{1/2} ∥. \end{matrix} (10)

Notice that the monotonicity of M_{(p)}(X) then gives that if X is q-concave for some finite q, then for all finite p there is a constant C_{p} so that the inequality

(𝔼 ∥ ∑_{n=1}^{N} ε_{n}x_{n}∥^{p})^{1/p} ≤ C_{p} (𝔼 ∥ ∑_{n=1}^{N} ε_{n}x_{n}∥) holds; this is a special case of the Kahane-Khintchine inequality that will be discussed in section 8.

Inequality (10) implies that if {x_{n}}_{n=1}^{∞} is an unconditional basic sequence in a Banach lattice which is p-concave, then the norm ∥ ∑_{n=1}^{N} α_{n}x_{n}∥ of a linear com-

bination is equivalent to the norm of the square function ∥(∑_{n=1}^{N} |α_{n}x_{n}|^{2})^{1/2}∥; more precisely,

\begin{matrix}A_{1}K^{−1} ∥(∑_{n=1}^{N} |α_{n}x_{n}|^{2})^{1/2}∥ & ≤ ∥ ∑_{n=1}^{N} α_{n}x_{n}∥ \\ & ≤ M_{(p)}(X)B_{p}K ∥(∑_{n=1}^{N} |α_{n}x_{n}|^{2})^{1/2}∥ \end{matrix} (11)

where K is the unconditional constant of {x_{n}}_{n=1}^{∞}. Since the left side of (11) is valid in any lattice, a duality argument yields that if {x_{n}}_{n=1}^{∞} is an unconditional basis for a complemented subspace of any Banach lattice, then ∥ ∑_{n=1}^{N} α_{n}x_{n}∥ is equivalent to ∥(∑_{n=1}^{N} |α_{n}x_{n}|^{2})^{1/2}∥. We write the equivalence in a slightly different way in which it is easier to keep track of the constants. Suppose that T : Y → X, S : X → Y are operators with ST = I_{Y} (so that T S is a projection from X onto SY), Y has a monotonely unconditional basis {y_{n}}_{n=1}^{∞}, and X is a Banach lattice. Then:

\begin{matrix}A_{1}∥T ∥^{−1} ∥(∑_{n=1}^{N} |α_{n}T y_{n}|^{2})^{1/2}∥ & ≤ ∥ ∑_{n=1}^{N} α_{n}y_{n}∥ \\ & ≤ A_{1}^{−1}∥S∥ ∥(∑_{n=1}^{N} |α_{n}T y_{n}|^{2})^{1/2}∥ . \end{matrix} (12)

Let {y_{n}^{∗}}_{n=1}^{∞} be the functionals in Y^{∗} biorthogonal to {y_{n}}_{n=1}^{∞}. By composing S with (contractive) projections onto the span of initial segments of {y_{n}}_{n=1}^{∞}, it can be assumed that Y is finite dimensional, in which case {y_{n}^{∗}} is a monotonely unconditional basis for Y^{∗}. To prove the right inequality in (12), given ∑_{n=1}^{N} α_{n}y_{n}, choose a norm one functional ∑_{n=1}^{N} β_{n}y_{n}^{∗} in Y^{∗} so that ∥ ∑_{n=1}^{N} α_{n}y_{n}∥ = ∑_{n=1}^{N} β_{n}y_{n}^{∗}(∑_{n=1}^{N} α_{n}y_{n}). Using (6) and the left inequality in (10), we then get

∥ ∑_{n=1}^{N} α_{n}y_{n}∥ = ∑_{n=1}^{N} β_{n}α_{n}S^{∗}y_{n}^{∗}(T y_{n})
≤ ∥ (∑_{n=1}^{N} |β_{n}S^{∗}y_{n}^{∗}|^{2})^{1/2} ∥∥ (∑_{n=1}^{N} |α_{n}T y_{n}|^{2})^{1/2} ∥
≤ A_{1}^{−1}𝔼∥ ∑_{n=1}^{N} ε_{n}β_{n}S^{∗}y_{n}^{∗}∥∥ (∑_{n=1}^{N} |α_{n}T y_{n}|^{2})^{1/2} ∥
≤ A_{1}^{−1}∥S^{∗}∥∥ ∑_{n=1}^{N} β_{n}y_{n}^{∗}∥ ∥(∑_{n=1}^{N} |α_{n}T y_{n}|^{2})^{1/2}∥ ,

which is the right side of (12).

Continuing in the situation considered in the previous paragraph, we analyze further the case when X is a C(K) space and ∥y_{n}∥ = 1 for all n. Since {y_{n}}_{n=1}^{∞} is unconditionally monotone, we have max_{n} |α_{n}| ≤ ∥ ∑_{n=1}^{N} α_{n}y_{n}∥. From (12), Hölder’s inequality, and properties of the supremum norm we get

∥ ∑_{n=1}^{N} α_{n}y_{n}∥ ≤ A_{1}^{−1}∥S∥∥(max_{n} |α_{n}T y_{n}|^{1/2})(∑_{n=1}^{N} |α_{n}T y_{n}|)^{1/2}∥

≤ A_{1}^{−1}∥S∥ (max_{n} |α_{n}|∥T y_{n}∥)^{1/2} max_{±} ∥ ∑_{n=1}^{N} ±α_{n}T y_{n}∥^{1/2}

≤ A_{1}^{−1}∥S∥∥T ∥∥ ∑_{n=1}^{N} α_{n}y_{n}∥^{1/2} max_{n} |α_{n}|^{1/2}

so that

∥ ∑_{n=1}^{N} α_{n}y_{n}∥ ≤ (A_{1}^{−1}∥T ∥∥S∥)^{2} max_{n} |α_{n}|.

Thus {y_{n}}_{n=1}^{∞} is K-equivalent (K := (A_{1}^{−1}∥T ∥∥S∥)^{2}) to the unit vector basis of c_{0}. Therefore, a seminormalized unconditional basis for a complemented subspace of a C(K) space is equivalent to the unit vector basis for c_{0}. By duality one obtains that a seminormalized unconditional basis for a complemented subspace of an L_{1}(μ) space is equivalent to the unit vector basis for ℓ_{1}. Since C(0, 1) and L_{1}(0, 1) contain subspaces isomorphic to ℓ_{2} while c_{0} and ℓ_{1} do not, this gives another proof that neither C(0, 1) nor L_{1}(0, 1) has an unconditional basis.

By using a p-convexification procedure, it is possible to build a scale of Banach lattices starting with a lattice X in a manner analogous to how the scale of L_{p}(μ) spaces is constructed from L_{1}(μ). For simplicity, we assume that X is a lattice of μ-measurable functions (for the general case see [15, p. 53]). For 1 < p < ∞, the p-convexification of X is the space X^{(p)} of all μ-measurable functions x for which |x|^{p}sign (x) is in X. This is easily seen to be a Banach lattice under the norm ∥x∥_{X^{(p)}} := ∥ |x|^{p}∥_{X}^{1/p}. The space X^{(p)} is p-convex with M^{(p)}(X^{(p)}) = 1. More generally, if X is r-convex and s-concave, then X^{(p)} is pr-convex and ps-concave with moduli M^{(pr)}(X^{(p)}) ≤ M^{(r)}(X)^{1/p} and M_{(ps)}(X^{(p)}) ≤ M_{(s)}(X)^{1/p}. If the lattice X is itself p-convex, one can also define the p-concavification of X to be the space X_{(p)} of all μ-measurable functions x for which |x|^{1/p}sign (x) is in X. The expression ∥ |x|^{1/p}∥^{p} is a norm if M_{(p)}(X) = 1, which, as we mentioned, can be achieved with an equivalent renorming of X.

6 Geometry of the norm

One interesting and fruitful line of research, dating from the early days of Banach space theory, has been to relate analytic properties of a Banach space to various geometrical conditions on the norm of the Banach space. The simplest example of such a condition is that of strict convexity: A Banach space X is strictly convex provided that the equality ∥x + y∥ = ∥x∥ + ∥y∥ with y nonzero implies that x is a scalar multiple of y. This just means that the unit sphere of X does not contain any line segment or that every point of the unit sphere is an extreme point of the unit ball. Related to strict convexity is smoothness: A smooth point of the unit ball of X is a point x in the unit sphere for which there is only one norm one linear functional which achieves its norm at x. The space X is called smooth provided every point in its unit sphere is a smooth point of the unit ball.

The spaces L_{p}(μ), 1 < p < ∞, are strictly convex and smooth, while the spaces L_{1}(μ) and C(K) are neither strictly convex nor smooth except in the trivial case when they are one dimensional.

It is easy to check that if X^{∗} is strictly convex [respectively, smooth], then X is smooth [respectively, strictly convex]. The converse is true for reflexive spaces but not in general.

In order to understand the analytic meaning that a point x in the unit sphere of X is a smooth point, consider any other vector y in X and a linear functional x^{∗} on X with ∥x^{∗}∥ = x^{∗}(x) = 1. Then for any t > 0, 1 + tx^{∗}(y) = x^{∗}(x + ty) ≤ ∥x + ty∥. That is,

x^{∗}(y) ≤ \frac{∥x + ty∥ − ∥x∥}{t}.

By using the triangle inequality one checks that the function \frac{∥x+ty∥−∥x∥}{t} is an increasing function of t on (0, ∞) and thus

x^{∗}(y) ≤ lim_{t→0+} \frac{∥x + ty∥ − ∥x∥}{t}.

Similarly,

x^{∗}(y) ≥ lim_{t→0+} \frac{∥x − ty∥ − ∥x∥}{−t}.

Thus if these two limits coincide, the value of x^{∗}(y) is uniquely determined. On the other hand, if these two limits differ, it follows from the Hahn-Banach theorem that for any λ satisfying

lim_{t→0+} \frac{∥x − ty∥ − ∥x∥}{−t} ≤ λ ≤ lim_{t→0+} \frac{∥x + ty∥ − ∥x∥}{t}

there is a linear functional x^{∗} for which ∥x^{∗}∥ = x^{∗}(x) = 1 and x^{∗}(y) = λ. Consequently, x is a smooth point if and only if ∥x + ty∥ + ∥x − ty∥ − 2 = o(t) for every y in X.

Quantitative versions of strict convexity and smoothness are of particular importance in Banach space theory. A Banach space X is said to be uniformly convex provided for every ϵ > 0 there exists δ_{X}(ϵ) = δ(ϵ) > 0 so that

sup{∥(x + y)/2∥ : ∥x∥ = ∥y∥ = 1; ∥x − y∥ = ϵ} = 1 − δ(ϵ).

The function δ_{X}(ϵ) is called the modulus of convexity of X. It is geometrically obvious (and even true) that in the definition of δ(ϵ), the equalities can be replaced by the inequalities ∥x∥ ≤ 1, ∥y∥ ≤ 1, ∥x − y∥ ≥ ϵ without changing the value of δ(ϵ).

One of the first theorems to relate the geometry of the norm to linear topological properties is the following. A uniformly convex space is reflexive. In order to see this, we may assume as well that X is separable. If X is not reflexive, then for any λ > 0 there is x^{∗∗} in the unit sphere of X^{∗∗} whose distance to X exceeds 1−λ. Let A be a countable subset of the unit sphere of X^{∗} which determines the norm of Y ≡ span X ∪ {x^{∗∗}}; that is, for each y^{∗∗} in Y, ∥y^{∗∗}∥ = sup{y^{∗∗}(x^{∗}) : x^{∗} ∈ A}. Let {x_{n}}_{n=1}^{∞} be a sequence in the unit sphere of X so that x^{∗}(x_{n}) → x^{∗∗}(x^{∗}) for every x^{∗} in A. Then lim_{n,m→∞} ∥x_{n} + x_{m}∥ = 2, while for every n, lim inf_{m→∞} ∥x_{n} − x_{m}∥ ≥ 1 − λ. This argument shows that for nonreflexive X, δ_{X}(ϵ) = 0 for every 0 < ϵ < 1 and in particular that X is not uniformly convex.

A Banach space X is said to be uniformly smooth if the function ρ_{X}(τ) = ρ(τ) = sup\{\frac{∥x+τy∥+∥x−τy∥}{2} − 1 : ∥x∥ = ∥y∥ = 1\} satisfies ρ(τ) = o(τ) as τ → 0. Again one checks easily that a uniformly smooth Banach space is reflexive.

There is a complete duality between uniform convexity and uniform smoothness. The space X is uniformly convex if and only if X^{∗} is uniformly smooth. This follows from a formula which connects the moduli δ_{X} and ρ_{X^{∗}}:

ρ_{X^{∗}}(τ) = sup{τϵ/2 − δ_{X}(ϵ) : 0 ≤ ϵ ≤ 2}.

To prove this identity, let x and y be in the unit sphere of X with ∥x − y∥ = ϵ and pick x^{∗}, y^{∗} in the unit sphere of X^{∗} so that x^{∗}(x + y) = ∥x + y∥ and y^{∗}(x − y) = ∥x − y∥. Then

2ρ_{X^{∗}}(τ) ≥ ∥x^{∗} + τy^{∗}∥ + ∥x^{∗} − τy^{∗}∥ − 2
≥ (x^{∗} + τy^{∗})(x) + (x^{∗} − τy^{∗})(y) − 2
= x^{∗}(x + y) + τy^{∗}(x − y) − 2 = ∥x + y∥ + τϵ − 2.

This shows that ρ_{X^{∗}}(τ) ≥ sup{τϵ/2 − δ_{X}(ϵ) : 0 ≤ ϵ ≤ 2}. Conversely, given τ > 0 and x^{∗}, y^{∗} in the unit sphere of X^{∗}, take x and y in the unit sphere of X so that (x^{∗} + τy^{∗})(x) = ∥x^{∗} + τy^{∗}∥ and (x^{∗} − τy^{∗})(x) = ∥x^{∗} − τy^{∗}∥ (if one wishes to avoid reflexivity at this stage one can choose x and y where the norms of the functionals are almost attained).Then

\begin{aligned}∥x^{∗} + τy^{∗}∥ & +∥x^{∗} − τy^{∗}∥ ≤ x^{∗}(x + y) + τy^{∗}(x − y) \\ & ≤ ∥x + y∥ + τ∥x − y∥ = 2δ_{X}(ϵ) + ϵτ, \end{aligned}

where ϵ = ∥x − y∥. This proves the reverse inequality.

The moduli of convexity and smoothness of a Hilbert space H can be easily computed from the parallelogram identity:

δ_{H}(ϵ) = 1 − \sqrt{1 − \frac{ϵ^{2}}{4}} = \frac{ϵ^{2}}{8} + O(ϵ^{4})

ρ_{H}(τ) = \sqrt{1 + τ^{2}} − 1 = \frac{τ^{2}}{2} + O(τ^{4})

Hilbert spaces are the “most uniformly convex” and “most uniformly smooth” spaces in the sense that if X is any space whose dimension is at least two, then δ_{H}(ϵ) ≥ δ_{X}(ϵ) and ρ_{H}(τ) ≤ ρ_{X}(τ). For infinite dimensional X this is an immediate consequence of Dvoretzky’s theorem, to be discussed in section 8, but a direct 2-dimensional geometrical argument gives the general case.

The moduli of convexity and smoothness of the L_{p} spaces can be computed exactly. It is somewhat easier to describe their asymptotic behavior near zero, and this is what really matters for Banach space theory. The result is

δ_{L_{p}}(ϵ) = \{ \begin{matrix}(p − 1)\frac{ϵ^{2}}{8} + o(ϵ^{2}), & if 1 < p ≤ 2 \\ \frac{ϵ^{p}}{p2^{p}} , & if 2 ≤ p < ∞. \end{matrix}

ρ_{L_{p}}(τ) = \{ \begin{matrix}\frac{τ^{p}}{p} + o(τ^{p}), & if 1 < p ≤ 2 \\ (p − 1)\frac{τ^{2}}{2} + o(τ^{2}), & if 2 ≤ p < ∞. \end{matrix}

This result can be extended to Banach lattices. If X is a Banach lattice which is p-convex with 1 < p ≤ 2 and q-concave with 2 ≤ q < ∞ and M^{(p)}(X) = 1 = M_{(q)}(X), then δ_{X}(ϵ) ≥ ϵ^{q}/C and ρ_{X}(τ) ≤ Cτ^{p} for some constant C = C(p, q) > 0 (see [15, 1f.1]).

Since it is easier to do analysis on a Banach space which has a norm with good geometric properties than on a general space, it is important to know which Banach spaces can be equivalently renormed so as to become strictly convex or smooth or uniformly convex or uniformly smooth, and it is useful to have in these last cases moduli which are as good as possible. The simplest, but nevertheless useful, renorming technique is as follows. Suppose that T is an injective operator from a Banach space X into a strictly convex Banach space Y. Then it is easy to check that |||x||| := ∥x∥ + ∥T x∥ is an equivalent strictly convex norm on X. Moreover, if T is an isomorphism and Y is uniformly convex, then |||·||| is an equivalent uniformly convex norm on X. Now if X is separable, then there is an injective operator T from X into ℓ_{2}, so ∥x∥_{1} = ∥x∥ + ∥T x∥ is an equivalent strictly convex norm on X. Also, there is an operator S from ℓ_{2} into X with dense range, so S^{∗} is an injective operator from X^{∗} into ℓ_{2}^{∗} = ℓ_{2} and ∥x^{∗}∥_{2} = ∥x^{∗}∥ + ∥S^{∗}x^{∗}∥ defines an equivalent strictly convex norm on X^{∗}. Since the adjoint operator S^{∗} is weak^{∗} to weak continuous, ∥·∥_{2} is dual to a (necessarily smooth) norm ∥·∥_{2} on X. If X is smooth and T is an injective operator from X into ℓ_{2} then (∥x∥^{2} + ∥T x∥_{2}^{2})^{1/2} is an equivalent norm which is both strictly convex and smooth. Hence every separable Banach space has an equivalent norm which is both strictly convex and smooth.

For certain nonseparable spaces; in particular, ℓ_{∞}(Γ) with Γ uncountable (see [6, Ch. II.7]), there may be no equivalent strictly convex or smooth norm.

If we are interested in obtaining a uniformly convex or smooth equivalent norm we have to restrict attention to reflexive spaces. However, not every reflexive space can be so renormed. For example, if (∑_{n=1}^{∞} ℓ_{1}^{n})_{2} had an equivalent uniformly convex norm ∥ · ∥, and ∥ · ∥_{n} denotes the restriction of ∥ · ∥to the nth coordinate space ℓ_{1}^{n}, then the expression |||x||| := lim ∥x∥_{n} (where “lim” is interpreted to be a limit over some free ultrafilter on ℕ or a Banach limit or the limit along an appropriate subsequence) defines an equivalent norm on the finitely supported vectors in ℓ_{1} which extends uniquely to an equivalent uniformly convex norm on ℓ_{1}, but this is impossible.

There is a characterization of those spaces on which there is an equivalent uniformly convex norm. These spaces, called superreflexive spaces, are discussed in section 9. The superreflexive spaces are also the class of spaces on which there is an equivalent uniformly smooth norm. (These deep facts are discussed in [29].) If a space X has an equivalent uniformly convex norm |||·|||, then the equivalent uniformly convex norms are dense in the metric space of equivalent norms (considered as bounded functions on the unit sphere of X). Take, for example, ∥ · ∥ + ϵ |||·|||. The uniformly convex equivalent norms also form a G_{δ} set since those whose modulus of convexity at 1/n is positive forms an open set. Thus the equivalent uniformly convex norms on X is a dense G_{δ} in the space of equivalent norms on X. Since, as we mentioned, X^{∗} also admits an equivalent uniformly convex norm, it follows by duality that the equivalent uniformly smooth norms on X is also a dense G_{δ} in the space of equivalent norms on X, hence so is the family of equivalent norms which are simultaneously uniformly convex and uniformly smooth.

In section 8 it is pointed out that an infinite dimensional L_{p}(μ) space cannot be equivalently renormed so as to have a better modulus of convexity or smoothness than that of the natural norm.

There has long been a desire to describe reflexivity geometrically (that is, to show that a space is reflexive if and only if there is an equivalent norm on the space that satisfies some geometrical condition), and the notion of uniform convexity pushed in that direction. This problem was recently solved (at least for separable spaces) and is discussed in [29].

Early in a first course in functional analysis a student learns that a Banach space is reflexive if and only if its closed unit ball is weakly compact. There is a beautiful and useful characterization of weak compactness, called James’ theorem, which does not explicitly involve the weak topology: A nonempty closed convex subset C of a Banach space X is weakly compact if and only if every x^{∗} in X^{∗} attains its maximum on C. The only if part is of course trivial. For the hard direction see [11, Th. 79] or [26] for an accessible proof when X is separable. This theorem says that on a nonreflexive space X there exist linear functionals which do not attain their norm on the unit ball of X. Nevertheless, the functionals which attain their norm on the unit ball is a rich set: The Bishop-Phelps theorem says: Let C be a nonempty closed bounded subset of a Banach space X. Then the functionals which attain their maximum on C is (norm) dense in X^{∗}. This theorem, which is the starting point of the theory of optimization on Banach spaces, has many extensions and applications (see [25]).

We outline the proof of the Bishop-Phelps theorem. First note that if f is a continuous bounded function on a complete metric space (U, d), then there is, for every ϵ > 0, a point u_{0} in U so that f(u) ≤ f(u_{0}) + ϵd(u, u_{0}) for every u in U. Indeed, define a partial order on U by u << v if f(u) ≤ f(v) − ϵd(u, v). The boundedness of f and completeness of U imply via Zornication that there is a maximal element u_{0} in this partial order. This u_{0} evidently has the desired property.

Suppose now that C is a nonempty closed bounded convex subset of X, ϵ > 0, and x^{∗} is in X^{∗}. By the remark above there is a point x_{0} in C so that x^{∗}(x_{0}) ≥ x^{∗}(x)−ϵ∥x−x_{0}∥ for all x in C. Consider two convex sets K_{1} and K_{2} in X ⊕ℝ:

\begin{matrix}K_{1} := \{(x, t) : x ∈ C; x^{∗}(x) ≥ t\} \\ K_{2} := \{(x, t) : x ∈ X; t ≥ x^{∗}(x_{0}) + ϵ∥x − x_{0}∥\}. \end{matrix}

The set K_{2} has a nonempty interior which is disjoint from K_{1}, so the separation theorem gives a nonzero point (u^{∗}, α) in X^{∗} ⊕ ℝ and a β so that u^{∗}(x) + αt ≥ β for (x, t) in K_{1} and u^{∗}(x) + αt ≤ β for (x, t) in K_{2}. Clearly α must be negative, so by normalizing we may assume that α = −1. Since (x_{0}, x^{∗}(x_{0})) is in K_{1} ∩ K_{2}, it follows that β = u^{∗}(x_{0}) − x^{∗}(x_{0}). By considering K_{1} we see that x^{∗} − u^{∗} attains its maximum on C at x_{0} and by considering K_{2} we deduce that ∥u^{∗}∥ ≤ ϵ.

The proof of the Bishop-Phelps theorem uses heavily the order in the real field. In fact, the Bishop-Phelps theorem does not hold in the complex case. There is a nonempty bounded closed convex set C in a complex Banach space X so that for every x^{∗} in X^{∗}, |x^{∗}| does not attain its maximum on C. This is discussed in [26].

We conclude this section by mentioning the geometric meaning of the Radon-Nikodým property or RNP, an analytical concept that will be discussed in section 7. A slice of a closed bounded convex set C is a set of the form S(C, x^{∗}, α) = {x ∈ C : x^{∗}(x) ≥ sup_{y∈C} x^{∗}(y) − α} with x^{∗} in X^{∗} and α > 0. C is called dentable provided that for each ϵ > 0 there is a slice of C which has diameter smaller than ϵ. In [8, Th. V.3.9] and [3, Th. 5.8] it is shown that X has the RNP if and only if every nonempty closed bounded convex subset of X is dentable. Here we show how dentability can be used to derive other geometric conditions, in particular, a version of the Krein-Milman theorem valid for noncompact subsets of a space which has the RNP. If X has the RNP, then every closed bounded convex subset C of X is the closed convex hull of its extreme points. Recall that a face of a convex set C is a nonempty (necessarily convex) subset F such that if λx + (1 − λ)y ∈ F with x, y in C and 0 < λ < 1 then x and y are in F. Extreme points are faces consisting of a single point. From the definition of slice it is obvious that if |(x^{∗} − y^{∗})(y)| ≤ δ for all y in C then S(C, y^{∗}, α − 2δ) ⊂ S(C, x^{∗}, α) as long as α > 2δ. We first show that C has an extreme point. Take any slice S(C, x^{∗}, α) of C whose diameter is less than one. By the observation above and the Bishop-Phelps theorem, there is a y^{∗} arbitrarily near x^{∗} which attains a maximum on C and so that the points P_{1} in C at which y^{∗} attains its maximum is contained in the slice S(C, x^{∗}, α) and thus has diameter less than one. Evidently P_{1} is a closed face of C and of course is dentable since X has the RNP. By induction we get a sequence {P_{n}}_{n=1}^{∞} so that P_{n+1} is a face of P_{n} and diam P_{n} ≤ 1/n. Since a face of a face is a face, all the P_{n}’s are faces of C and their intersection is an extreme point of C which is in the slice S(C, x^{∗}, α). To finish the proof, let K be the closed convex hull of the extreme points of C. If K were properly contained in C, then the separation theorem and the Bishop-Phelps theorem would yield a functional x^{∗} which attains a maximum on C and so that the set of points P in C at which x^{∗} attains its maximum is disjoint from K. By what we have proved P has an extreme point which is a fortiori an extreme point of C, which contradicts the definition of K.

It is open whether this extreme point property for every nonempty closed bounded convex subset of a space actually characterizes spaces with the RNP.

The statement about extreme points is however valid in a stronger form which trivially implies that the dentability condition is equivalent to the RNP. A point x in C is called a exposed point of C if there is x^{∗} in X^{∗} which attains its maximum on C exactly at the single point x. If also the diameter of the slice S(C, x^{∗}, α) tends to zero as α → 0+, then x is called a strongly exposed point of C. For example, every point on the unit sphere of a uniformly convex space is a strongly exposed point of the unit ball. The stronger statement to which we alluded above is: Every nonempty closed bounded convex subset of a space with the RNP is the closed convex hull of its strongly exposed points.

7 Analysis in Banach spaces

In this section we describe the basic facts concerning integration and differentiation in Banach spaces as well as the connection of these topics to convexity. Our “point of view” is to take as known the scalar theory but not assume that the reader has any familiarity with the vector valued theory. Much of what we treat in this section is contained either in [8] or in [11], [12]. Almost everything is in [3].

Let (Ω, μ) be a complete σ-finite measure space and X a Banach space. A function from Ω → X of the form ∑_{i=1}^{n} x_{i}1_{A_{i}} with each A_{i} a measurable subset of Ω is called a simple function. A function f : Ω → X is called strongly measurable or just measurable if it is the limit almost everywhere of a sequence of simple functions. A function f is called scalarly measurable provided x^{∗}f is measurable for each x^{∗} in X^{∗}. A function f is measurable if and only if it is scalarly measurable and the range of f is essentially separable, which means that for some set A of measure zero, f[Ω ∼ A] is separable. The only if assertion is clear. To verify the if part, we may assume that X is separable and select a sequence {x_{n}^{∗}}_{n=1}^{∞} in the unit ball of X^{∗} so that ∥x∥ = sup_{n} x_{n}^{∗}(x) for every x in X. Thus for each fixed x in X, the function ∥f − x∥ is a measurable real valued function on Ω. Since X is separable, given ϵ > 0 there is a sequence {x_{n}}_{n=1}^{∞} in X and a sequence {A_{n}}_{n=1}^{∞} of disjoint measurable subsets of Ω so that ∥f − ∑_{n} x_{n}1_{A_{n}}∥ < ϵ. That is, f can be approximated uniformly by functions taking only countably many values and having all level sets measurable and hence f is the limit a.e. of a sequence of simple functions.

We next define the notion of the Bochner integral of a vector valued function. This is the “strongest” of the various vector valued integrals which have been considered and is the one most useful for the topics treated in this Handbook.

If f is a simple function supported on a set of finite measure, define ∫ f dμ = ∑_{x∈X} μ[f = x]x. This is of course a finite sum. It is easy to see directly, and also follows from the scalar case by composing with linear functionals, that the integral is linear on this class of simple functions. If f is the a.e. limit of a sequence {f_{n}}_{n=1}^{∞} of simple functions supported on sets of finite measure and lim_{n,m} ∫ ∥f_{n} − f_{m}∥ dμ = 0, then ∫ f dμ is defined to be lim_{n} ∫ f_{n} dμ. It is easy to check that ∫ f dμ does not depend on the particular sequence {f_{n}}_{n=1}^{∞} and that ∫ f dμ exists if and only if f is measurable and ∫ ∥f∥ dμ < ∞. That the integral has the expected properties either follows from the scalar case by composing with linear functionals or is easy to check directly; see [8, pp. 44-52]. In particular, ∥ ∫ f dμ∥ ≤ ∫ ∥f∥ dμ and T ∫ f dμ = ∫ T f dμ whenever f is integrable and T is an operator.

That the usual differentiability properties hold for the Bochner integral can be deduced from the scalar theorems even without recalling the proofs in that case. Suppose that f is a Lebesgue integrable X valued function on ℝ^{n}. Then for a.e. u in ℝ^{n}, we have

lim_{r→0} m(B(0, r))^{−1} ∫_{B(u,r)} ∥f(v) − f(u)∥ dv = 0 (13)

and thus also f(u) = lim_{r→0} m(B(0, r))^{−1} ∫_{B(u,r)} f(v) dv (here integration is with respect to Lebesgue measure m on ℝ^{n}). Indeed, we may assume that X is separable and {x_{n}}_{n=1}^{∞} is dense in X. Then by the scalar theorem,

lim_{r→0} m(B(0, r))^{−1} ∫_{B(u,r)} ∥f(v) − x_{n}∥ dv = ∥f(u) − x_{n}∥

for a.e. u and every n. For a point u for which this holds for every n we have

lim sup_{r→0} m(B(0, r))^{−1} ∫_{B(u,r)} ∥f(v) − f(u)∥ dv ≤
lim sup_{r→0} m(B(0, r))^{−1} ∫_{B(u,r)} ∥f(v) − x_{n}∥ + ∥x_{n} − f(u)∥ dv
= 2∥f(u) − x_{n}∥.

Since {x_{n}}_{n=1}^{∞} is dense in X we get (13), as desired. In particular, if f : ℝ → X is integrable and F (t) = ∫_{0}^{t} f then F′(s) = f(s) a.e.

The Banach space L_{p}(μ, X), 1 ≤ p < ∞, is defined to be the space of all measurable X valued functions for which ∥f∥_{p} := (∫ ∥f∥^{p} dμ)^{1/p} < ∞ (with the usual modification when p = ∞). The simple functions which are supported on sets of finite measure are dense in L_{p}(μ, X), 1 ≤ p < ∞. When μ is counting measure, L_{p}(μ, X) = ℓ_{p}(X) = (∑ X)_{p}.

We mention two important notions of differentiability for a function f from a Banach space X into a Banach space Y. f is said to be Gâteaux differentiable or G-differentiable at a point x_{0} in X if there is an operator T from X into Y so that for every u in X lim_{t→0}(f(x_{0} + tu) − f(x_{0}))/t = T u. If this limit exists uniformly with respect to u in the unit sphere of X the function f is said to be Fréchet differentiable or F-differentiable at x_{0}. That is, f is F -differentiable at x_{0} provided f(x_{0} + u) = f(x_{0}) + T u + o(∥u∥). The operator T is unique if it exists and is called the G or F derivative of f at x_{0} and is denoted by D_{f}(x_{0}). If f is F -differentiable at x_{0} it is continuous there and standard two dimensional examples show that G-differentiability does not imply continuity. However, if f is Lipschitz and X is finite dimensional, then G-differentiability is easily seen to imply F -differentiability. This is false for infinite dimensional spaces and this is one of the main causes for the difficulty of the subject of derivatives in infinite dimensions. For example, the function f(u) = sin u is an everywhere G-differentiable Lipschitz map from X = L_{2}(0, 1) into itself with D_{f}(u)(v) = cos u·v, but f is nowhere F -differentiable (for example, at u_{0} = 0, sin 1_{(0,t)} − 1_{(0,t)} = (sin 1 − 1)1_{(0,t)} is not o(∥1_{(0,t)}∥_{2}) as t → 0).

In general, familiar finite dimensional theorems (chain rule, inverse function theorem, implicit function theorem, etc.) go over to F -derivatives with the same proofs, while for G-derivatives one must proceed with care. On the other hand, the existence theorems for G-derivatives are quite satisfactory while F -derivatives often do not exist and the question when they exist leads to difficult problems. See [39] for a discussion of this topic.

A direct application of the scalar mean value theorem yields that if the Gderivative D_{f}(x) exists in a neighborhood of x_{0} and is continuous at x_{0}, then it is actually an F -derivative. Thus the notion of C^{1} function is the same in the context of G or F derivatives. For C^{1} functions we can speak of the second derivative in either the G or F sense. The second derivative is then an operator from X into the space of operators from X to Y or, alternatively, a bounded bilinear map from X ⊕ X to Y. Similarly, one can speak of C^{n} or C^{∞} functions. One can define analytic functions when X and Y are complex spaces as those C^{∞} functions f for which the Taylor expansion converges in some neighborhood and represents f there. A useful fact that is proved in elementary textbooks (such as [19, Ch. 3]) is that a function f from the complex plane into Y is analytic if y^{∗}f is analytic for each y^{∗} in Y^{∗}.

One important Banach space theory property that comes from differentiation theory is the Radon-Nikodým property or RNP. One way to define this notion is the following: The space X has the RNP if every Lipschitz function from ℝ into X is differentiable a.e. Here we do not need to specify Gâteaux or Fréchet differentiability since the notions obviously coincide for all functions from ℝ. The RNP is usually defined by the requirement that the Radon-Nikodým theorem holds for X valued measures of finite total variation (see for example [8]). We shall discuss that aspect and other equivalences of the RNP later in this section. Here just note that a Banach space valued Lipschitz function f on [0, 1] has separable range, so its a.e. derivative f′, if it exists, is also separably valued and thus is a measurable function since x^{∗}f is measurable for each x^{∗} in X^{∗}.

First let us get a feeling for which spaces have the RNP. It is clear from the definition that a subspace of a space with the RNP has the RNP and that a space all of whose separable subspaces have the RNP has the RNP as well. The mapping t ↦ 1_{(0,t)} from (0, 1) into L_{1}(0, 1) shows that L_{1}(0, 1) fails the RNP. Also c_{0} fails the RNP. This is seen by considering the function f : ℝ → c_{0}⊕_{∞}c_{0} defined by f(t) = {∫_{0}^{t} sin ns ds}_{n=1}^{∞} ⊕ {∫_{0}^{t} cos ns ds}_{n=1}^{∞}. That f maps into c_{0} follows from the Riemann-Lebesgue lemma but the only possible candidate for a derivative is {sin ns}_{n=1}^{∞} ⊕ {cos ns}_{n=1}^{∞} which is not in c_{0} for any s.

Separable conjugate spaces have the RNP and thus all reflexive spaces have the RNP (incidentally, this yields another proof of the fact mentioned in section 3 that L_{1}(0, 1) does not embed into a separable conjugate space). To see this, let f be a Lipschitz function into a separable conjugate space Z^{∗} and let {z_{n}}_{n=1}^{∞} be dense in Z. For all n, the scalar Lipschitz function f(t)(z_{n}) is differentiable for a.e. t. At a point t_{0} where all of these functions are differentiable, f(t_{0})(z) is differentiable for every z in Z (observe that h^{−1} (f(t_{0} + h) − f(t_{0})) (z) − k^{−1} (f(t_{0} + k) − f(t_{0})) (z) → 0 as h, k → 0 because the difference quotient is uniformly bounded since f is Lipschitz and tends to zero on the dense set {z_{n}}_{n=1}^{∞}). From this we conclude that the limit g(t) := lim_{h→0} h^{−1} (f(t + h) − f(t)) exists a.e., but the limit is only in the weak^{∗} sense. This is all that can be said using just the separability of Z. However, since Z^{∗} is separable, we deduce that g is measurable (g has separable range and ∥g(t) − z^{∗}∥ is clearly measurable for every z^{∗} in Z^{∗}; this is all that was used in the proof of measurability in the beginning of this section). Also g is bounded by the Lipschitz constant of f, so the Bochner integral G(s) := ∫_{0}^{s} g(t) dt is well defined for all s. Evaluating both sides at an arbitrary z in Z we see from the scalar theory that G(s)(z) = (f(s) − f(0)) (z) so that f(s) = G(s) + f(0) and hence by what we proved on the differentiation of the integral we conclude that f′(s) = G′(s) = g(s) a.e. also in the sense that ∥h^{−1} (f(s + h) − f(s)) − g(s)∥ → 0 as h → 0.

Thus every subspace of a separable conjugate space has the RNP and a space with the RNP has no subspace isomorphic to c_{0} or L_{1}(0, 1). These facts give useful criteria for the RNP and in certain cases, such as spaces with an unconditional basis, allow a complete determination whether a space has the RNP. However, there do exist separable spaces with the RNP which do not embed into a separable conjugate space and there are spaces failing the RNP which do not contain isomorphic copies of either c_{0} or L_{1}(0, 1) (see [3, section 3.4]).

Before stating the usual definition of the RNP we need to introduce the notion of what is termed in [8] a “countably additive vector measure on a σ-algebra”. If X is a Banach space, an X valued measure τ is a countably additive function from a σ-algebra into X. For this to make sense, the series ∑ τ(A_{n}) must converge unconditionally for each sequence {A_{n}}_{n=1}^{∞} of disjoint sets in the σ-algebra. One defines the total variation |τ| of an X valued measure by |τ|(A) = sup ∑_{n} ∥τ(A_{n})∥, where the supremum is over all partitions of A into finitely many (or countably many; it is the same) disjoint sets in the σ-algebra (Ω, B). The measure τ is said to be of finite variation provided |τ|(Ω) < ∞. If τ is of finite variation, then |τ| is a finite scalar measure. For example, if Ω = ℕ, B is the collection of all subsets of ℕ, and {x_{n}}_{n=1}^{∞} is a sequence in the Banach space X, then the assignment τ{n} := x_{n} extends to an X valued measure on B if and only if ∑_{n} x_{n} converges unconditionally. The measure is then of finite variation if and only if ∑_{n} ∥x_{n}∥ < ∞. An example of an L_{p}(0, 1), 1 ≤ p < ∞, valued measure is gotten by taking the Lebesgue measurable subsets of [0, 1] and defining τ(A) = 1_{A}. This measure has finite variation only if p = 1.

The theory of X valued measures and more general vector measures is exposed in [8]. In addition to their importance for the geometry of Banach spaces, vector measures are centrally important for spectral theory; the ones encountered there typically do not have finite variation.

An X valued measure τ is absolutely continuous with respect to the finite scalar measure μ provided τ(A) = 0 for every set of μ measure zero. Because μ is finite, this is equivalent to saying that for every ϵ > 0 there is δ > 0 so that ∥τ(A)∥ < ϵ whenever μ(A) < δ (see [8, p. 10]). If τ is an X valued measure of finite variation it is clear that τ is absolutely continuous with respect to its total variation |τ| and even satisfies (14) below (with μ := |τ|).

An X valued measure τ is differentiable with respect to a scalar measure μ provided that there is an X valued measurable function g so that τ(A) = ∫_{A} g dμ for every measurable set A. We say that the Radon-Nikodým theorem holds in X provided that if τ is an X valued measure of finite variation and τ is absolutely continuous with respect to a finite scalar measure μ, then τ is differentiable with respect to μ. If X satisfies this condition only for all separable finite scalar measures, we say that the separable Radon-Nikodým theorem holds in X (a measure μ is called separable provided L_{1}(μ) is separable). The usual definition is that a Banach space X has the RNP provided the Radon-Nikodým theorem holds in X and this is equivalent to saying that the separable Radon-Nikodým theorem holds in X (see [8, Ch. III]). Later we prove this equivalence for separable X, but first we show a general space X has the RNP if and only if the separable Radon-Nikodým theorem holds in X.

Suppose that the separable Radon-Nikodým theorem holds in X and let f : [0, 1] → X be a Lipschitz function. One defines a linear mapping T from the step functions on [0, 1] into X by setting T 1_{[a,b]} := f(b)−f(a) for a subinterval of [0, 1] and extending linearly. Since f is a Lipschitz function, the mapping T is continuous when the step functions are given the L_{1}(0, 1) norm, and hence T uniquely extends to an operator (also denoted by T) from L_{1}(0, 1) into X. The assignment τ(A) := T 1_{A} obviously defines an X valued measure of finite variation which is absolutely continuous with respect to Lebesgue measure m, so we get an X valued measurable function g on [0, 1] for which τ(A) = ∫_{A} g dm for every Lebesgue measurable subset of [0, 1]. In particular, f(t) = ∫_{0}^{t} g dm + f(0) and thus by what we proved in the beginning of this section f′(t) exists a.e. on [0, 1] (and is equal to g(t)).

Suppose that X has the RNP. To see that the separable Radon-Nikodým theorem holds in X, suppose first that the scalar “control measure” μ is Lebesgue measure on [0, 1] and that the X valued measure τ satisfies

∥τ(A)∥ ≤ μ(A) (14)

for all Lebesgue measurable subsets of [0, 1]. Define f(t) = τ[0, t]. Evidently f is Lipschitz and so by assumption is differentiable a.e. on [0, 1] and f′ is measurable. That τ(A) = ∫_{A} f′ dμ for all Lebesgue measurable sets now follows from the scalar theory by composing with linear functionals.

By the measure isomorphism theorem already used in section 4, we get that if the finite control measure μ is separable and the X valued measure τ satisfies (14), then τ is differentiable with respect to μ.

Now suppose that ν is a finite separable measure and τ is an X valued measure which is absolutely continuous with respect to ν. Then μ := |τ| is a finite scalar measure which is absolutely continuous with respect to ν, so by the scalar Radon-Nikodým theorem there is a ν-measurable function f ≥ 0 so that μ(A) = ∫_{A} f dν for every ν-measurable set A. Of course, μ is then also a separable measure and, as we have already remarked, τ satisfies (14), so from what we already have proved there is an X valued μ-measurable function g so that τ(A) = ∫_{A} g dμ for every μ-measurable set A. Then f · g is ν-measurable and τ(A) = ∫_{A} f · g dν for every ν-measurable set A.

Observe that the simple argument reducing the study of an X valued measure which is absolutely continuous with respect to a finite control measure to the case where the X valued measure satisfies (14) yields another characterization of the RNP; more precisely, that the [separable] Radon-Nikodým theorem holds in X if and only if for each operator T from an L_{1}(μ) space with μ finite [and separable] into X there is an X valued measurable function g so that T f = ∫ fg dμ for all f in L_{1}(μ). From this it is easy to see that if X is separable and the separable Radon-Nikodým theorem holds in X then the Radon-Nikodým theorem holds in X. Indeed, let μ be a finite measure on a σ-algebra B and let T : L_{1}(μ) → X be an operator. Since X is separable there is a sequence {x_{n}^{∗}}_{n=1}^{∞} in X^{∗} which separates the points of X. Let A be a countably generated sub σ-algebra of B so that all the L_{∞}(μ) functions T^{∗}x_{n}^{∗} are A-measurable. Since {x_{n}^{∗}}_{n=1}^{∞} separates points of X, T f = T 𝔼(f|A) for each f in L_{1}(μ). The restriction of μ to A is a separable measure since A is countably generated. Thus we get an X valued A-measurable function g so that T f = ∫ f · g dμ for each A-measurable function f in L_{1}(μ). But then for a general f in L_{1}(μ) we have T f = T 𝔼(f|A) = ∫ 𝔼(f|A)g dμ = ∫ f ·g dμ since g is A-measurable.

One of many places where the RNP arises naturally is in the study of vector valued L_{p} spaces. There is a natural isometric identification of L_{p^{∗}}(μ, X^{∗}), 1/p+1/p^{∗} = 1, with a subspace of L_{p}(μ, X)^{∗}, and for 1 ≤ p < ∞, L_{p}(μ, X)^{∗} = L_{p^{∗}}(μ, X^{∗}) for all finite (or σ-finite) measures μ if and only if X^{∗} has the RNP (see [8, Ch. IV]).

There are other important analytic characterizations of spaces with the RNP in terms of martingales. In particular, the RNP spaces are exactly those Banach spaces in which the martingale convergence theorem is valid in the sense that X has the RNP if and only if every L_{1} bounded X valued martingale converges a.e. (see [8, Ch. V]).

It turns out that in many places where one might assume reflexivity in order to use weak compactness of the unit ball it suffices to assume that the space has the RNP.

We now discuss the differentiability of (real valued) convex continuous functions on a Banach space X. Part of the importance of this topic derives from the fact that the norm is a convex continuous function and differentiability of the norm is intrinsically related to its smoothness. The most elementary reference for the differentiability of convex functions and related topics is probably [11, Ch. 5].

An easy consequence of the definition is that a locally bounded convex function is continuous and even locally Lipschitz. By using the separation theorem in X ⊕ ℝ it follows that whenever f is convex and continuous in a neighborhood of a point x_{0} the set (called the subdifferential of f) ∂_{f}(x_{0}) := {x^{∗} ∈ X^{∗} : x^{∗}(x − x_{0}) ≤ f(x) − f(x_{0}) for all x ∈ X} is nonempty. From the theory of convex functions on ℝ we know that for each u the right and left derivatives of the function t ↦ f(x_{0} + tu) exist at t = 0. These one-sided derivatives agree for every u (that is, all directional derivatives exist at x_{0}) if and only if ∂_{f}(x_{0}) is a single point which is then necessarily the G-derivative of f at x_{0}. Consequently f is G-differentiable at x_{0} if and only if for every u, f(x_{0} +tu)+ f(x_{0} − tu) − 2f(x_{0}) = o(t) as t → 0.

By considering f(x) = ∥x∥ we recover the fact mentioned in section 6 that the norm is G-differentiable at x_{0} in the unit sphere of X if and only if x_{0} is a smooth point of the unit ball of X. It also follows that f is F -differentiable at x_{0} if and only if f(x_{0} + u) + f(x_{0} − u) − 2f(x_{0}) = o(∥u∥) as ∥u∥ → 0. If a convex function f is F -differentiable in a neighborhood of x_{0} then D_{f}(x) is continuous there; that is, F -differentiability of a convex function on an open set implies that it is C^{1} there. Indeed, suppose that x_{n} → x_{0} and set w^{∗} := D_{f}(x_{0}); u_{n}^{∗} := D_{f}(x_{n}). Given ϵ > 0 there is δ > 0 so that f(x_{0} + y) − f(x_{0}) − w^{∗}(y) ≤ ϵ∥y∥ for ∥y∥ ≤ δ. Pick y_{n} with ∥y_{n}∥ = δ and (u_{n}^{∗} − w^{∗})(y_{n}) ≥ (δ/2)∥u_{n}^{∗} − w^{∗}∥ for all n. Then since u_{n}^{∗} = D_{f}(x_{n}), we have by the convexity of f that

ϵδ + w^{∗}(y_{n}) + f(x_{0}) ≥ f(x_{0} + y_{n}) ≥ u_{n}^{∗}(y_{n} − x_{n} + x_{0}) + f(x_{n})

or

\begin{aligned}(δ/2) & ∥u_{n}^{∗} − w^{∗}∥ ≤ (u_{n}^{∗} − w^{∗})(y_{n}) \\ & ≤ ϵδ − f(x_{n}) + f(x_{0}) + ∥u_{n}^{∗}∥ · ∥x_{n} − x_{0}∥. \end{aligned}

Since ∥u_{n}^{∗}∥ is bounded by the local Lipschitz constant of f, we conclude that lim sup_{n} ∥w^{∗} − u_{n}^{∗}∥ ≤ 2ϵ.

Specializing again to f(x) = ∥x∥, we see that the norm is G-differentiable at every nonzero x if and only if X is smooth (the norm is clearly never differentiable at zero); that is, for each x in the unit sphere of X there is a unique unit functional j_{x} of norm one which achieves its norm at x. The norm is then F -differentiable at every nonzero x if and only if the duality mapping j is a norm-to-norm continuous mapping. It follows from this and the Bishop-Phelps theorem that if ∥·∥ is F -differentiable away from zero then the density character of X^{∗} is equal to that of X. In particular, X^{∗} is separable if X is.

The space X is uniformly smooth if the norm is uniformly F -differentiable on the unit sphere; that is, if the limit lim_{t→0} t^{−1}(∥x+tu∥−∥x∥) exists uniformly in both x and u on the unit sphere.

The classical Gâteaux differentiability theorem for convex functions says: A continuous convex function f on a separable Banach space X is Gdifferentiable on a dense G_{δ} set. Indeed, if {x_{n}}_{n=1}^{∞} is dense in X then the set of G-differentiability of f is the set ∩_{n,m}G_{n,m}, where G_{n,m} is the set of points x in X for which there exists δ > 0 so that f(x + δu_{n}) + f(x − δu_{n}) − 2f(x) ≤ δ/m. It is readily verified that each G_{n,m} is open and dense.

For Fréchet differentiability, the situation is much different even for norms. The norm of ℓ_{1} is G-differentiable at any point all of whose coordinates are nonzero but is nowhere F -differentiable. This is typical in the sense that if X is separable and X^{∗} is nonseparable then X admits an equivalent norm that is nowhere F -differentiable (see [11, Th. 106]). On the other hand: If X^{∗} is separable then every convex function f on X is F -differentiable on a dense G_{δ}. That the set of points of F -differentiability is a G_{δ} is easy to check (and for this no separability assumption is needed); it is equal to ∩_{n}G_{n} where G_{n} is the set of points x in X for which there exists δ > 0 so that sup_{∥u∥≤1} f(x + δu) + f(x − δu) − 2f(x) ≤ δ/n; the G_{n}’s are obviously open. That each G_{n} is dense is not as obvious; see [11, Th. 114].

We now consider the differentiability of Lipschitz (or just locally Lipschitz) functions. The theorem of Lebesgue on the a.e. differentiability of Lipschitz functions on the line has a classical extension (called Rademacher’s theorem) which says that every Lipschitz mapping between finite dimensional spaces is differentiable a.e. The proof goes over to show that the finite dimensional assumption on the range of the Lipschitz map can be replaced by the assumption that the range has the RNP. The main problem weakening the assumption on the domain space in Rademacher’s theorem is that one must come up with an appropriate notion of a.e. or negligible set in an infinite dimensional space. It turns out that there are several nonequivalent ways to do this and we shall discuss one of them.

A Borel subset A of a separable Banach space X is said to be Haar null provided there exists a probability measure μ on the Borel subsets of X so that μ(x + A) = 0 for every x in X. The Haar null sets form a σ-ring and they coincide with the usual Borel sets of Lebesgue measure zero if X is finite dimensional. In an infinite dimensional space X, every compact set A is Haar null since there is a direction so that every line in this direction cuts A in a set of linear measure zero and thus μ can be any probability measure supported on a line in this direction which is equivalent to linear Lebesgue measure.

Once one has the notion of Haar null set, the classical proof of Rademacher’s theorem can be modified to prove: Every locally Lipschitz mapping from a separable Banach space into a Banach space with the RNP is Gdifferentiable off a Haar null set. For a fuller discussion and a discussion of other notions of negligible set, see [39] or [3].

This theorem does not hold for F -differentiability. Even ℓ_{2} has an equivalent norm whose set of points of F -differentiability is a Haar null set (see [3, Ex. 6.46]). For locally Lipschitz functions one deep and useful result is: Every real valued locally Lipschitz function on a space whose dual is separable is F -differentiable at a dense set of points (again, see [39] or [3]). It is not known if this theorem remains true for Lipschitz functions taking values in the plane.

8 Finite dimensional Banach spaces

Since all finite dimensional spaces of the same dimension over the same scalar field are mutually isomorphic, for results on finite dimensional spaces to be meaningful they must be of a quantitative nature. The notion of Banach-Mazur distance is of central importance in this context. Evaluating or even estimating the distance between spaces is often hard since it generally is quite difficult to find an operator T for which ∥T ∥ ∥T^{−1}∥ is minimal or close to minimal.

We illustrate the computation of the Banach-Mazur distance by evaluating (or in some cases just giving the order of magnitude) the quantity d(ℓ_{p}^{n}, ℓ_{r}^{n}). While relatively easy, even in this simple situation it is by no means trivial to calculate or even closely estimate the distance when p and r are on different sides of two. The topic of Banach-Mazur distances between finite dimensional space is treated in many more involved situations in [20].

Denote the formal identity mapping from ℓ_{p}^{n} to ℓ_{r}^{n} by I_{p,r}. It is trivial to check that ∥I_{p,r}∥ = 1 when p ≤ r and ∥I_{p,r}∥ = n^{1/r−1/p} when p > r. Consequently, if 1 ≤ p < r ≤ ∞, then d(ℓ_{p}^{n}, ℓ_{r}^{n}) ≤ n^{1/p−1/r}. The simplest and most important case occurs when p (or r) is two. Either by induction from the case n = 2, where the following equality is just the parallelogram law, or by using the orthonormality of a Rademacher sequence and taking the vectors x_{i} to be in e.g. L_{2}(0, 1) and exchanging the order of integration over [0, 1] with the expectation, one sees that vectors x_{1}, …, x_{n} in a Hilbert space satisfy the identity

𝔼 ∥ ∑_{i=1}^{n} ε_{i}x_{i}∥_{2}^{2} = ∑_{i=1}^{n} ∥x_{i}∥_{2}^{2}. (15)

Suppose that 2 < r ≤ ∞ and T : ℓ_{r}^{n} → ℓ_{2}^{n} is an isomorphism normalized to satisfy ∥T^{−1}∥ = 1 (so that ∥T x∥_{2} ≥ ∥x∥_{r} for all x). Denoting as usual the unit vectors basis as {e_{i}}, we see from (15) that n ≤ ∑_{i=1}^{n} ∥T e_{i}∥_{2}^{2} = 𝔼 ∥ ∑_{i=1}^{n} ε_{i}T e_{i}∥_{2}^{2} ≤ max_{±} ∥T (∑_{i=1}^{n} ±e_{i}) ∥_{2}^{2} ≤ ∥T ∥^{2}n^{2/r}, which is to say that ∥T ∥ ≥ n^{1/2−1/r}. From the equality d(ℓ_{2}^{n}, ℓ_{r}^{n}) = n^{1/2−1/r} and the (multiplicative) triangle inequality for the Banach-Mazur distance one gets that d(ℓ_{p}^{n}, ℓ_{r}^{n}) = n^{1/p−1/r} if 2 ≤ p ≤ r ≤ ∞. By duality (or by repeating essentially the same argument) one gets the same is true if 1 ≤ p ≤ r ≤ 2.

It turns out that when p < 2 < r the identity operator I_{p,r} does not give even a good approximation to the Banach-Mazur distance. For the upper estimate it is enough to work with complex scalars. This is because complex ℓ_{p}^{n} as a real space is isometric to the real space (∑_{1}^{n} ℓ_{2}^{2})_{p} which for all p has distance at most \sqrt{2} to ℓ_{p}^{2n}. Using d_{ℂ}(·, ·) for the complex Banach-Mazur distance and d_{ℝ}(·, ·) for the real distance, we thus have for even dimensions d_{ℝ}(ℓ_{p}^{2n}, ℓ_{r}^{2n}) ≤ 2d_{ℂ}(ℓ_{p}^{n}, ℓ_{r}^{n}) and for odd dimensions d_{ℝ}(ℓ_{p}^{2n+1}, ℓ_{r}^{2n+1}) ≤ 1 + 2d_{ℂ}(ℓ_{p}^{n}, ℓ_{r}^{n}).

The reason that the complex case is simpler is that there exists for every n a complex unitary matrix V_{n} of order n all of whose entries have absolute value 1/\sqrt{n} (take for V_{n} the matrix whose j,k entry is n^{−1/2} exp(2πikj/n)).

Denote the norm of an operator ∥T ∥ from ℓ_{p}^{n} to ℓ_{r}^{n} by ∥T ∥_{p,r}. Using the obvious identity ∥T ∥ = max_{i} ∥T e_{i}∥ for operators T with domain ℓ_{1}^{n}, we have that ∥V_{n}∥_{1,∞} = n^{−1/2}. By considering V_{n}^{∗} as the composition I_{2,1}V_{n}^{∗}I_{∞,2} we see that ∥V_{n}^{∗}∥_{∞,1} ≤ n. Since V_{n}^{∗} = V_{n}^{−1}, it follows that d(ℓ_{1}^{n}, ℓ_{∞}^{n}) ≤ n^{−1/2}. For the corresponding upper estimate for the distance for general 1 ≤ p < 2 < r ≤ ∞, we need to use interpolation, which will be discussed in section 11 (but we use here only the most classical theorem). First we do the case r = p^{∗} , where 1/p + 1/p^{∗} = 1. From the identities ∥V_{n}∥_{1,∞} = n^{−1/2} and ∥V_{n}∥_{2,2} = 1 interpolation gives us the inequality ∥V_{n}∥_{p,p^{∗}} ≤ n^{1/2−1/p}. Since V_{n}^{∗} = V_{n}^{−1} we see that

\begin{aligned}d(ℓ_{p}^{n}, ℓ_{p^{∗}}^{n}) & ≤ ∥V_{n}∥_{p,p^{∗}}∥V_{n}^{∗}∥_{p^{∗},p} ≤ n^{1/2−1/p}∥I∥_{2,p}∥V_{n}^{∗}∥_{2,2}∥I∥_{p^{∗},2} \\ & = n^{1/2−1/p}n^{1/p−1/2} · 1 · n^{1/p−1/2} = n^{1/p−1/2}. \end{aligned}

In the general case 1 ≤ p < 2 < r ≤ ∞, by replacing the pair {p, r} by {r^{∗}, p^{∗}} if necessary, we can assume that p^{∗} ≤ r. Then the triangle inequality for the Banach-Mazur distance gives us d(ℓ_{p}^{n}, ℓ_{r}^{n}) ≤ n^{1/p−1/2}d(ℓ_{p^{∗}}^{n}, ℓ_{r}^{n}) = n^{1/2−1/r}. So for arbitrary 1 ≤ p < 2 < r ≤ ∞ we have

d(ℓ_{p}^{n}, ℓ_{r}^{n}) ≤ max{n^{1/p−1/2}, n^{1/2−1/r}} (16)

in the case of complex scalars and something slightly worse in the case of real scalars (e.g. the same except that the right side of (16) is multiplied by three).

To show that the estimate (16) is accurate (up to a constant) one uses again an argument of averaging against a Rademacher sequence. The argument works in both the real and complex case. The inequality needed, called the cotype 2 inequality for L_{p}, will be discussed later in this section. It states that for 1 ≤ p ≤ 2, any collection x_{1}, …, x_{n} of vectors in an L_{p}(μ) space satisfies the inequality

(𝔼 ∥ ∑_{i=1}^{n} ε_{i}x_{i}∥_{p}^{2})^{1/2} ≥ A_{p} (∑_{i=1}^{n} ∥x_{i}∥_{p}^{2})^{1/2} , (17)

where the expectation is with respect to the Rademacher sequence {ε_{n}}_{n=1}^{∞}. Applying (17) to the image of the unit vector basis of ℓ_{r}^{n}, we get for r > 2 that d(ℓ_{r}^{n}, E) ≥ A_{p}n^{1/2−1/r} for every subspace E of an L_{p}(μ) space. Since A_{p} ≥ A_{1}, this shows that the upper estimate of (16) is precise up to an absolute constant.

Since an n dimensional Banach space can be identified with ℝ^{n} [or ℂ^{n}] with some symmetric [or balanced] closed bounded convex body as the unit ball, it is natural that geometric arguments often occur in the study of finite dimensional spaces. The notion of volume is of particular importance in this regard. Suppose that X and Y are both n dimensional Banach spaces, which we regard as ℝ^{n} [or ℂ^{n}] with some norm, and α(·) is a norm on the space B(X, Y) of operators from X to Y. The natural algebraic dual of B(X, Y) can be identified with B(Y, X) under the trace duality < S, T >:= trace ST and we denote by α^{∗}(·) that norm on B(Y, X) which makes (B(Y, X), α^{∗}) the dual of (B(X, Y), α) under trace duality. By compactness of the unit ball of a finite dimensional space, there exists among all operators T : X → Y with α(T) ≤ 1 one which maximizes the volume of T B_{X}. Lewis’ lemma states: If vol (T B_{X}) achieves a maximum subject to the constraint α(T) ≤ 1 at T = T_{0}, then T_{0} is invertible and α^{∗}(T_{0}^{−1}) = n. This means n^{−1}T_{0}^{−1} is a norm one functional on (B(X, Y), α) which achieves its norm at T_{0}.

Before giving the proof of Lewis’ lemma, we mention one consequence of the idea of maximizing volume, Auerbach’s lemma, which does not use Lewis’ lemma: If Y is an n dimensional Banach space then there is a basis {x_{k}, x_{k}^{∗}}_{k=1}^{n} so that for each k, ∥x_{k}∥ = 1 = ∥x_{k}^{∗}∥. We take X = ℓ_{1}^{n} and let α(·) be the operator norm: Having gotten T_{0} : ℓ_{1}^{n} → Y to maximize the volume of the image of the ℓ_{1}^{n} ball as above, it is obvious that the basis {x_{k}, x_{k}^{∗}}_{k=1}^{n} works, where x_{k} := T e_{k}, x_{k}^{∗}(x) := det(T_{0})^{−1} det(T_{k}), and T_{k} : ℓ_{1}^{n} → Y is defined by letting T_{k}e_{i} be x_{i} when i ≠ k and T_{k}e_{k} = x.

We turn to the proof of Lewis’ lemma. Since the volume of T C for any measurable set is a constant multiple of | det(T)|vol (C), for any operator S : X → Y we have

|det (\frac{T_{0} + S}{α(T_{0} + S)})| ≤ | det(T_{0})|. (18)

Certainly T_{0} must be invertible, so by dividing (18) by | det(T_{0})| we can rewrite (18) as

| det(I_{X} + T_{0}^{−1}S)| ≤ α(T_{0} + S)^{n}. (19)

Since α(T_{0}) ≤ 1 we get from (19) that for all ϵ > 0 and all operators S : X → Y:

| det(I_{X} + ϵT_{0}^{−1}S)| ≤ (1 + ϵα(S))^{n}. (20)

Obviously det(I_{X} +ϵT) = 1+ϵ(trace T)+o(ϵ) as ϵ → 0 (that is, the trace functional is the derivative of the determinant at the identity operator), so from (20) we conclude that trace T_{0}^{−1}S ≤ nα(S) for all operators S : X → Y which in view of the trace duality between α(·) and α^{∗}(·) means that α^{∗}(T_{0}^{−1}) ≤ n.

The first application of Lewis’ lemma comes from applying it when α(·) is the operator norm. (This case, known before the lemma, see e.g. [20, p. 137], in fact motivated the discovery of the lemma.) The norm N (·) on B(Y, X) which is dual to the operator norm on B(X, Y) for finite dimensional X and Y is called the nuclear norm. To identify the nuclear norm on B(Y, X), observe first that if ∥y^{∗}∥ = 1 = ∥x∥ with y^{∗} in Y^{∗} and x in X, then N (y^{∗} ⊗ x) = 1, where y^{∗} ⊗ x is the rank one operator from Y to X defined by (y^{∗} ⊗ x)(z) = y^{∗}(z)x. Next note that for any operator T : X → Y,

∥T ∥ = sup{y^{∗}(T x) : ∥y^{∗}∥ = 1 = ∥x∥}
= sup{trace (y^{∗} ⊗ x)T : N (y^{∗} ⊗ x) = 1}.

Applying the bipolar theorem we conclude that the unit ball of (B(Y, X), N (·)) is the convex hull of the closed set W := {y^{∗} ⊗ x : ∥y^{∗}∥ = 1 = ∥x∥}. From this description of the unit ball of (B(Y, X), N (·)) one obtains that for any S in B(Y, X), N (S) = min ∑_{k=1}^{N} ∥y_{k}^{∗}∥ · ∥x_{k}∥, where the minimum is over all representations S = ∑_{k=1}^{N} y_{k}^{∗} ⊗ x_{k}. Caratheodory’s theorem says that the minimum occurs already with N at most one plus the dimension of B(Y, X) as a vector space over the reals; that is, N ≤ n^{2} + 1 in the case of real scalars and N ≤ 2n^{2} + 1 when the scalars are complex. (By being slighty more careful before applying Caratheodory’s theorem, one obtains the best estimates N ≤ n^{2} and N ≤ 2n^{2}; see [20, 8.6]). Having done these preliminaries, we can state a nice geometric reformulation of the case α(·) = ∥ · ∥ in Lewis’ lemma. Again identify both spaces X and Y with ℝ^{n} or ℂ^{n}. When one convex body contains another, a point x is called a contact point of the bodies if it is in the intersection of their boundaries (so if the bodies are unit balls associated with two norms, a contact point is a point in the intersection of the two unit spheres). Assume that B_{X} ⊂ B_{Y} and vol (B_{X}) ≥ vol (T B_{X}) for every operator T on ℝ^{n} [ℂ^{n}] for which T B_{X} ⊂ B_{Y}. Then there exist contact points x_{1}, …, x_{N} of B_{X} and B_{Y} and contact points x_{1}^{∗}, …, x_{N}^{∗} of B_{X^{∗}} and B_{Y^{∗}} and c_{k} ≥ 0 so that I = ∑_{k=1}^{N} c_{k}x_{k}^{∗} ⊗ x_{k}. Also, N ≤ n^{2} in the real case and N ≤ 2n^{2} in the complex case.

The most important applications of Lewis’ lemma occur when X = ℓ_{2}^{n}. In this case observe that the operator T_{0} is unique up to a unitary operator on ℓ_{2}^{n}. Indeed, suppose that T_{1} is another operator from ℓ_{2}^{n} for which α(T_{1}) = 1 and α^{∗}(T_{1}^{−1}) = n. By preceding T_{1} with a unitary we can assume that T_{0}^{−1}T_{1} is a positive operator D with eigenvalues α_{1} ≥ … ≥ α_{n} > 0. Then ∑_{k=1}^{n} α_{k} = trace D ≤ α^{∗}(T_{0}^{−1})α(T_{1}) ≤ n while ∑_{k=1}^{n} 1/α_{k} = trace D^{−1} ≤ α^{∗}(T_{1}^{−1})α(T_{0}) ≤ n, so that α_{k} = 1 for all k. Specializing to the case of the operator norm, we obtain a classical result called John’s theorem: Let X be ℝ^{n} or ℂ^{n} under some norm ∥ · ∥. Then B_{X} contains a unique ellipsoid of maximal volume. This ellipsoid is the Euclidean ball B := B_{ℓ_{2}^{n}} if and only if there exist contact points {u_{k}}_{k=1}^{N} between B and B_{X} and c_{k} > 0 so that for all x in X,

x = ∑_{k=1}^{N} c_{k}⟨u_{k}, x⟩u_{k}. (21)

Moreover, N ≤ n(n + 1)/2 in the real case and N ≤ n^{2} in the complex case. To see that John’s theorem follows from the geometric version of Lewis’ lemma, write the identity on X as I_{X} = ∑_{k=1}^{N} c_{k}x_{k}^{∗} ⊗ x_{k} with the x_{k} contact points between B and B_{X}, the x_{k}^{∗} contact points between B and B_{X^{∗}}, and the c_{k} positive real numbers which sum to n. Here B_{X^{∗}} is identified with those y in X for which |⟨y, x⟩| ≤ 1 for all x in B_{X} and the duality pairing is given by the usual inner product. By taking traces we see that ⟨x_{k}^{∗}, x_{k}⟩ = 1 for all k and hence, since the Euclidean norm of x_{k}^{∗} and x_{k} are both one, that x_{k}^{∗} is the functional ⟨·, x_{k}⟩. This gives John’s theorem except for the improved estimate on N, which is easy but we omit (see [20, 8.6]).

An important consequence of John’s theorem for Banach space theory is: If X is n dimensional then d(X, ℓ_{2}^{n}) ≤ \sqrt{n}. Indeed, by (21) since ⟨u_{k}, ·⟩ is a norm one functional on X, we have for all x in X that ∥x∥_{2}^{2} = ⟨x, x⟩ = ∑_{k=1}^{N} c_{k}|⟨u_{k}, x⟩|^{2} ≤ ∑_{k=1}^{N} c_{k} = n. This says that B_{X} ⊂ \sqrt{n}B.

For general spaces of dimension n we thus have d(X, Y) ≤ d(X, ℓ_{2}^{n})d(ℓ_{2}^{n}, Y) ≤ n. It turns out that this estimate is precise up to a constant independent of n. There exist for each n spaces X_{n} and Y_{n} of dimension n so that inf d(X_{n}, Y_{n})/n > 0. If one puts 2n pairs of symmetric points on the unit sphere of ℓ_{2}^{n}, where the points are chosen independently and are distributed uniformly on that sphere, and then takes the symmetric convex hull of the union of these points with the unit vector basis, one obtains a unit ball for a (random) space. If one takes two of these random spaces, then with big probability (that is, with probability tending to one as n → ∞) the Banach-Mazur distance between the two spaces exceeds δn for some constant δ > 0 independent of n. Although this construction is easy to describe, the computations are delicate and the reader is referred to [20, 38.1] for details. The probabilistic approach has many other applications. For example, from the estimate d(X, ℓ_{2}^{n}) ≤ \sqrt{n} it follows immediately that an n dimensional space has a basis with basis constant at most \sqrt{n}. Spaces of the type X_{n} mentioned above exist (even with big probability) so that any basis in X_{n} has basis constant at least δ\sqrt{n}. A discussion of these examples and their applications to infinite dimensional theory is given in [36].

The maximal volume ellipsoid appears also in another geometrical result which is a weak version of the classical Dvoretzky-Rogers lemma (see [28]). Again let X be ℝ^{n} or ℂ^{n} under some norm for which the Euclidean ball is the ellipsoid of maximum volume contained in B_{X}. Then there exist [n/2] orthonormal vectors x_{k} so that ∥x_{k}∥_{X} ≥ 1/2 for all k. Rather than deduce this from John’s theorem, we prove it directly. Construct {x_{k}}_{k=1}^{n} inductively so that ∥x_{k+1}∥_{X} is maximized subject to the constraints that ∥x∥_{2} = 1 and x is orthogonal to x_{1}, …, x_{k}. Fix any k < n, set β = ∥x_{k}∥_{X}, and let X_{k} be the span of x_{1}, …, x_{k}. Define an ellipsoid D by D := {x + y : x ∈ X_{k}; y ∈ X_{k}^{⊥}; 2∥x∥_{2}^{2} + 2β^{2}∥y∥_{2}^{2} ≤ 1}. One checks easily that D ⊂ B_{X}. By comparing the volume of D to that of B one gets that 2^{−n/2}β^{−(n−k+1)} ≤ 1 which gives the estimate ∥x_{k}∥_{X} ≥ 2^{−n/2(n−k+1)}.

A geometric interpretation of the Dvoretzky-Rogers lemma is that whenever X has dimension 2n there is a subspace Y of dimension n so that in an appropriate coordinate system the unit ball of Y contains the Euclidean ball and is contained in twice the unit cube B_{ℓ_{∞}^{n}}. Since ℓ_{∞}^{n} is 1-injective, this implies that the identity I_{2,∞} : ℓ_{2}^{n} → ℓ_{∞}^{n} can be written as a product T_{1}T_{2} with ∥T_{2} : ℓ_{2}^{n} → X∥ = 1 and ∥T_{1} : X → ℓ_{∞}^{n}∥ ≤ 2.

A result which goes much further is Dvoretzky’s theorem, which says that every infinite dimensional Banach spaces contains for every n subspaces whose Banach-Mazur distance to ℓ_{2}^{n} is arbitrarily close to one. The finite dimensional quantitative version of Dvoretzky’s theorem from which the infinite dimensional statement above follows immediately says: For every k and ϵ > 0 there exists n_{0} = n_{0}(ϵ, k) so that if dim(X) ≥ n_{0} then X contains a subspace Y with d(Y, ℓ_{2}^{k}) < 1+ϵ. There are known good estimates on n_{0}(ϵ, k) (and even better estimates for special classes of spaces). For general spaces n_{0}(ϵ, k) ≤ exp(αk/ϵ^{2}) for some constant α. Dvoretzky’s theorem is treated in detail in several books, including [9, 19.1], [17, 4.3], [16, I.5.8]. An exposition of Dvoretzky’s theorem and related results is given in [28]. There are many proofs of Dvoretzky’s theorem and in most of them the Dvoretzky-Rogers lemma is the first step.

From Dvoretzky’s theorem and the technique for constructing basic sequences discussed in section 3 one gets in any infinite dimensional Banach space a basic sequence {x_{n}}_{n=1}^{∞} so that for each k, {x_{j}}_{j=2^{k}+1}^{2^{k+1}} is 1/k-equivalent to an orthonormal basis in a Hilbert space. An easy consequence of this is that for every square summable sequence {α_{n}}_{n=1}^{∞}, of scalars, every infinite dimensional Banach space contains an unconditionally convergent series ∑_{n} y_{n} such that for each n, ∥y_{n}∥ = |α_{n}|. One can also easily deduce this from the simple Dvoretzky-Rogers lemma; in fact, historically it was this application that motivated the discovery of the Dvoretzky-Rogers lemma.

A result related to Dvoretzky’s theorem is Krivine’s theorem: If {x_{n}}_{n=1}^{∞} is a basic sequence in a Banach space, then there exists 1 ≤ p ≤ ∞ so that for every k and every ϵ > 0 there is a block basis of {x_{n}}_{n=1}^{∞} of length k which is (1+ϵ)-equivalent to the unit vector basis of ℓ_{p}^{k}. This statement can be “localized” to obtain a statement about all sufficiently long finite basic sequences having a specified basis constant (or even biorthogonality constant). We do not do so here for two reasons: (1) The finite statement follows formally from the infinite statement via a localization technique that will be mentioned in section 9, and (2) Unlike the situation in Dvoretzky’s theorem, the quantitative estimates in the finite statement are so poor that there are no known consequences of them beyond what can be deduced formally from the infinite version of Krivine’s theorem stated above. For a proof of Krivine’s theorem see [16, Ch. 12] or [3, Ch. 12].

In the setting of Banach lattices, Krivine’s theorem yields that for every infinite dimensional Banach lattice there exists 1 ≤ p ≤ ∞ so that for every k and every ϵ > 0 there is a lattice isomorphism T from ℓ_{p}^{n} into X with ∥T ∥ ∥T^{−1}∥ < 1 + ϵ.

From Krivine’s theorem and the fact discussed in section 4 that ℓ_{2} embeds isometrically into L_{p}(0, 1) it is easy to deduce Dvoretzky’s theorem. This is not the recommended route to Dvoretzky’s theorem as it is difficult to navigate through Krivine’s theorem. More importantly, the tight quantitative estimates obtainable from direct proofs of Dvoretzky’s theorem have many applications.

If we apply Krivine’s theorem to a basic sequence which is equivalent to the unit vector basis for ℓ_{r} we see immediately that the only p that is obtainable is p = r. Thus by applying Krivine’s theorem to a disjoint sequence in an infinite dimensional L_{p}(μ) space we infer that any equivalent renorming of L_{p}(μ) contains for every k and ϵ > 0 a subspace whose Banach-Mazur distance to ℓ_{p}^{k} is less than 1 + ϵ. This implies that an infinite dimensional L_{p}(μ) space cannot be given an equivalent norm which has a better modulus of convexity or smoothness than its natural norm.

Two notions that are very important for both the finite dimensional and infinite dimensional theories are that of type and cotype. A Banach space X is said to have type p provided there is a constant C so that for every sequence x_{1}, …, x_{n} in X,

(𝔼 ∥ ∑_{i=1}^{n} ε_{i}x_{i}∥^{2})^{1/2} ≤ C (∑_{i=1}^{n} ∥x_{i}∥^{p})^{1/p} (22)

(with the usual modification for p = ∞). The expectation is with respect to the Rademacher sequence {ε_{n}}_{n=1}^{∞}. Similarly, X is said to have cotype q provided there is a constant C so that for every sequence x_{1}, …, x_{n} in X,

(∑_{i=1}^{n} ∥x_{i}∥^{q})^{1/q} ≤ C (𝔼 ∥ ∑_{i=1}^{n} ε_{i}x_{i}∥^{2})^{1/2} . (23)

The best constants in (22) and (23) are denoted by T_{p}(X) and C_{q}(X). By the parallelogram identity characterization of Hilbert space, a space X is isometric to a Hilbert space if and only if T_{2}(X) = 1 = C_{2}(X). Even a one dimensional space does not have type p for any p > 2 or cotype q for any q < 2. For every space X, T_{1}(X) = 1 and C_{∞}(X) = 1 by convexity of the norm. As functions of p and q, T_{p}(X) is nondecreasing and C_{p}(X) is nonincreasing. The inequalities T_{p}(X) ≤ d(X, Y)T_{p}(Y) and C_{q}(X) ≤ d(X, Y)C_{q}(Y) are evident, so for an n dimensional X, in the range 1 ≤ p ≤ 2 ≤ q ≤ ∞ the quantities T_{p}(X) and T_{q}(X) cannot exceed \sqrt{n} by the distance estimate of X to ℓ_{2}^{n}.

A Banach lattice X which is p-convex and q-concave has type p_{0} := p ∧ 2 and cotype q_{0} := q ∨ 2. The type assertion follows from (10) and monotonicity of M^{(t)} and M_{(s)}:

\begin{aligned}(𝔼 ∥ ∑_{n} ε_{n}x_{n}∥^{2})^{1/2} & ≤ M_{(q_{0})}(X)B_{q_{0}}∥ (∑_{n} |x_{n}|^{2})^{1/2} ∥ \\ & ≤ M_{(q)}(X)B_{q_{0}}M^{(p)}(X) (∑_{n} ∥x_{n}∥^{p_{0}})^{1/p_{0}} . \end{aligned}

For the lattice cotype assumption we need Khintchine’s inequality (1) as well as monotonicity of the convexity and concavity parameters:

(∑_{n} ∥x_{n}∥^{q_{0}})^{1/q_{0}} ≤ M_{(q_{0})}(X)∥ (∑_{n} |x_{n}|^{q_{0}})^{1/q_{0}} ∥

≤ M_{(q)}(X)∥ (∑_{n} |x_{n}|^{2})^{1/2} ∥ ≤ M_{(q)}(X)A_{p_{0}}^{−1}∥ (𝔼 | ∑_{n} ε_{n}x_{n}|^{p_{0}})^{1/p_{0}} ∥

≤ M_{(q)}(X)A_{p_{0}}^{−1}M^{(p_{0})}(X) (𝔼 ∥ ∑_{n} ε_{n}x_{n}∥^{p_{0}})^{1/p_{0}}
≤ M_{(q)}(X)A_{p_{0}}^{−1}M^{(p)}(X) (𝔼 ∥ ∑_{n} ε_{n}x_{n}∥^{2})^{1/2} .

Specializing to L_{p}(μ) spaces, we see that for 1 ≤ p ≤ 2, T_{p}(L_{p}) = 1 and C_{2}(L_{p}) ≤ A_{p}^{−1}(≤ A_{1}^{−1}), while for 2 ≤ p < ∞, T_{2}(L_{p}) ≤ B_{p} and C_{p}(L_{p}) = 1. Notice that this implies that for 1 ≤ r < p ≤ 2, ℓ_{r} is not isomorphic to a subspace of any L_{p}(μ) space.

By considering a disjoint sequence in the space, we see that an infinite dimensional L_{p}(μ) space with 1 ≤ p ≤ 2 cannot have type better than p. Also, since infinite dimensional L_{∞}(μ) spaces are universal for separable spaces, they do not have type p for any p > 1. Similarly, an infinite dimensional L_{q}(μ) space for 2 ≤ q ≤ ∞ does not have cotype smaller than q.

If {x_{n}}_{n=1}^{∞} is a seminormalized unconditionally basic sequence in L_{q}(μ), 2 < q < ∞, with μ a probability measure and inf ∥x_{n}∥_{2} > 0, then since L_{q}(μ) has type 2 and L_{2}(μ) has cotype 2 and ∥ · ∥_{2} ≤ ∥ · ∥_{q} we infer that {x_{n}}_{n=1}^{∞} is, in L_{q}(μ), equivalent to the unit vector basis of ℓ_{2}. Thus such a sequence cannot have dense linear span in L_{q}(μ). In particular, the trigonometric system is not unconditional in L_{q}(0, 1) for 2 < q ≤ ∞ and, by duality, also not in L_{p}(0, 1), 1 ≤ p < 2.

If {x_{n}}_{n=1}^{∞} is a seminormalized sequence in L_{q}(0, 1), 2 < q < ∞, and inf ∥x_{n}∥_{2} > 0, then we saw in section 4 that {x_{n}}_{n=1}^{∞} has a subsequence which is equivalent to the unit vector basis for ℓ_{q}. Since L_{p}(0, 1), 1 < p < ∞, has an unconditional basis (the Haar system), it follows from the discussion in section 3 that every seminormalized weakly null sequence in L_{p}(0, 1) has an unconditionally basis subsequence. Combining these comments with those in the previous paragraph, and using the fact that every separable subspace of L_{p}(μ) embeds isometrically into L_{p}(0, 1), we conclude that every seminormalized weakly null sequence in L_{q}(0, 1), 2 < q < ∞, has a subsequence which is equivalent to the unit vector basis for either ℓ_{2} or ℓ_{q}.

The second moment in the definitions of type and cotype can be replaced with any other moments (but of course then the constants change in (22) and (23)). This is a consequence of the Kahane-Khintchine inequality, which says that for every 0 < p < ∞ there are constants A_{p} and B_{p} so that for every sequence x_{1}, …, x_{n} in a Banach space,

A_{p} (𝔼 ∥ ∑_{i=1}^{n} ε_{i}x_{i}∥^{2})^{1/2} ≤ (𝔼 ∥ ∑_{i=1}^{n} ε_{i}x_{i}∥^{p})^{1/p} ≤ B_{p} (𝔼 ∥ ∑_{i=1}^{n} ε_{i}x_{i}∥^{2})^{1/2} (24)

Here the expectation is with respect to the Rademacher sequence {ε_{n}}_{n=1}^{∞}. The inequality (24) follows via extrapolation from the following hypercontractivity inequality: for every 2 < p < ∞ there is a constant σ_{p} > 0 so that for every sequence x, x_{1}, …, x_{n} in a Banach space,

(𝔼 ∥x + ∑_{i=1}^{n} ε_{i}σ_{p}x_{i}∥^{p})^{1/p} ≤ (𝔼 ∥x + ∑_{i=1}^{n} ε_{i}x_{i}∥^{2})^{1/2} . (25)

To prove (25), first note that there is σ_{p} > 0 so that for all 0 < t ≤ 1,

(\frac{|1 + σ_{p}t|^{p} + |1 − σ_{p}t|^{p}}{2} )^{1/p} ≤ (1 + t^{2})^{1/2}. (26)

This follows, for example, from L’Hôpital’s rule. The best constant is σ_{p} = (p − 1)^{−1/2} (see [15, 1.e.14]). Secondly, deduce a vector valued version of (26):

(𝔼 ∥x + ε_{1}σ_{p}y∥^{p})^{1/p} ≤ (𝔼 ∥x + ε_{1}y∥^{2})^{1/2}. (27)

One can assume that ∥x + y∥ + ∥x − y∥ = 2 with t := \frac{∥x+y∥−∥x−y∥}{2} ≥ 0. Then for ε = ±1,

∥x + σ_{p}εy∥ ≤ \frac{1 + σ_{p}}{2} ∥x + εy∥ + \frac{1 − σ_{p}}{2} ∥x − εy∥ = 1 + εσ_{p}t.

Therefore

(𝔼 ∥x + ε_{1}σ_{p}y∥^{p})^{1/p} ≤ (𝔼 (1 + σ_{p}ε_{1}t)^{p})^{1/p} ≤ \sqrt{1 + t^{2}} = (𝔼 ∥x + ε_{1}y∥^{2})^{1/2} .

Finally, iterate (27) to obtain (25). Formally, for k = 1, …, n define S_{k} := ∑_{i=1}^{k} ε_{i}x_{i}; we need to show that (𝔼∥x + σ_{p}S_{k}∥^{p})^{1/p} ≤ (𝔼∥x + S_{k}∥^{2})^{1/2}. For k = 1 this is (27). Assume that the desired inequality is true for k and let 𝔼_{k+1} be conditional expectation with respect to ε_{k+1} and 𝔼^{k} conditional expectation with respect to ε_{1}, …, ε_{k}. Then:

(𝔼∥x + σ_{p}S_{k+1} ∥^{p})^{2/p} = (𝔼^{k}𝔼_{k+1}∥(x + σ_{p}S_{k}) + ε_{k+1}σ_{p}x_{k+1}∥^{p})^{2/p}

≤ (𝔼^{k}(𝔼_{k+1}∥(x + σ_{p}S_{k}) + ε_{k+1}x_{k+1}∥^{2})^{p/2})^{2/p} by (27)

≤ 𝔼_{k+1} (𝔼^{k}∥(x + ε_{k+1}x_{k+1}) + σ_{p}S_{k}∥^{p})^{2/p} by Minkowski

≤ 𝔼_{k+1}𝔼^{k}∥(x + ε_{k+1}x_{k+1}) + S_{k}∥^{2} = 𝔼∥x + S_{k+1}∥^{2}.

The theory of type and cotype is quite extensive. In the rest of this section we mention without proof a sample of results in this theory. Expositions of this theory can be found in several books; in particular, [15], [16], [17], [20]; and see the article [38].

We already mentioned that a space isomorphic to a Hilbert space is of type 2 and cotype 2. The converse is true and this is one of the basic isomorphic characterizations of Hilbert space.

There is a connection between the theory of cotype and Dvoretzky’s theorem. If X is an n dimensional space then X has a subspace Y of dimension k ≥ n^{2/p}/N(p, C_{p}(X), ϵ) with d(Y, ℓ_{2}^{k}) ≤ 1 + ϵ. In particular, if X is a subspace of L_{1}(μ) then the k above is proportional to n, with a constant depending just on ϵ. See [17, 4.15] or [16, 5.3].

A connection between type and cotype theory and Krivine’s theorem is provided by the Maurey-Pisier theorem: If X is an infinite dimensional Banach space and p_{0} := sup{p : X has type p} and q_{0} := inf{q : X has cotype q}, then for any ϵ > 0 and any k there are subspaces Y_{k} and Z_{k} of X so that d(Y_{k}, ℓ_{p_{0}}^{k}) ≤ 1 + ϵ, and d(Z_{k}, ℓ_{p_{0}}^{k}) ≤ 1 + ϵ. A proof of this is given in [16, II.13.2].

The special cases p_{0} = 1 and q_{0} = ∞ of the Maurey-Pisier theorem are especially important (and much easier to prove; they do not require Krivine’s theorem). If X does not have type p for any p > 1, then X contains almost isometric copies of ℓ_{1}^{n} (the converse is true and obvious). Spaces which do not contain almost isometric copies of ℓ_{1}^{n} for all n arose first in the study of probability in Banach spaces and are sometimes called B-convex (see [34]).

Similarly, a space does not have finite cotype if and only if it contains for every n arbitrarily close copies of ℓ_{∞}^{n}. Since ℓ_{1}^{k} is a subspace of ℓ_{∞}^{2^{k}} (isometrically in the case of real scalars and up to constant in the complex case), we deduce that if X is B-convex then it has cotype q for some q < ∞. Since a uniformly convex space cannot have subspaces of arbitrary dimension which are arbitrarily close to ℓ_{1}^{n}, it follows that a uniformly convex space is B-convex; that is, has nontrivial type and cotype.

The relation of B-convexity to reflexivity is not so simple but has been clarified. A B-convex space need not be reflexive–there is even a nonreflexive space of type 2. Though nonreflexive Banach spaces need not contain almost isometric copies of ℓ_{1}^{n}, they do contain configurations of the following type for every n and ϵ > 0: norm one vectors x_{1}, …, x_{n} so that for every k < n, ∥x_{1} + … + x_{k} − (x_{k+1} + … + x_{n})∥ ≥ n − ϵ. In particular, taking n = 2 we see that: Every nonreflexive space contains real subspaces arbitrarily close to real ℓ_{1}^{2}. That is, a real space whose unit ball does not have a two dimensional section arbitrarily close to a square must be reflexive. See [2, 4.III] for proofs of these results.

We next discuss the duality theory of type and cotype. It is simple that if X has type p then X^{∗} has cotype p^{∗} and C_{p^{∗}}(X^{∗}) ≤ T_{p}(X). Indeed, given x_{1}^{∗}, …, x_{n}^{∗} in X^{∗} and ϵ > 0, take x_{1}, …, x_{n} in X so that ∑_{i=1}^{n} x_{i}^{∗}(x_{i}) ≥ (1 − ϵ) (∑_{i=1}^{n} ∥x_{i}^{∗}∥^{p^{∗}})^{1/p^{∗}} (∑_{i=1}^{n} ∥x_{i}∥^{p})^{1/p}. Using (4), we get

∑_{i=1}^{n} x_{i}^{∗}(x_{i}) = 𝔼 (∑_{i=1}^{n} ε_{i}x_{i}^{∗}) (∑_{i=1}^{n} ε_{i}x_{i})
≤ (𝔼 ∥ ∑_{i=1}^{n} ε_{i}x_{i}^{∗}∥^{2})^{1/2} (𝔼 ∥ ∑_{i=1}^{n} ε_{i}x_{i}∥^{2})^{1/2}
≤ T_{p}(X) (∑_{i=1}^{n} ∥x_{i}∥^{p})^{1/p} (𝔼 ∥ ∑_{i=1}^{n} ε_{i}x_{i}^{∗}∥^{2})^{1/2} .

In the opposite direction this does not work: L_{1} has cotype 2 but L_{∞} does not have type 2. Since L_{2}(X^{∗}) is a subspace of L_{2}(X)^{∗}, we have

(𝔼 ∥ ∑_{i=1}^{n} ε_{i}x_{i}^{∗}∥^{2})^{1/2} = sup{𝔼 (∑_{i=1}^{n} ε_{i}x_{i}^{∗}) f : ∥f∥_{L_{2}(X)} ≤ 1}.

The problem is the passage from a general f to a function of the form ∑_{i=1}^{n} ε_{i}x_{i}. This would be possible if the Rademacher projection P̃ := P ⊗I_{X} from L_{2}(X) into itself would be bounded, where P is the orthogonal projection onto the closed linear span of the Rademacher sequence {ε_{n}}_{n=1}^{∞} (a space X for which P̃ is bounded is said to be K-convex). P̃ is defined explicitly for f_{i} in L_{2}(μ) and x_{i} in X by P̃(∑_{i=1}^{n} f_{i}x_{i}) = ∑_{i=1}^{n}(P f_{i})x_{i}. It is a deep fact (see [16, II.14], [17, 2.4], and [38]) that: X is K-convex if and only if X is B-convex. Thus the presence of copies of ℓ_{1}^{n} is exactly the factor which hinders a clean duality between type and cotype. If X is B-convex and of cotype p^{∗} then X^{∗} is of type p.

For finite dimensional X it is important to estimate the norm of the Rademacher projection (called the K-convexity constant of X and usually denoted by K(X)). A useful estimate (which is valid also for isomorphs of infinite dimensional Hilbert space) is K(X) ≤ K(1 + log d(X, H)) for some constant K, where H is a Hilbert space with the same dimension as X. When dim(X) = n > 1, the distance estimate from John’s theorem gives that K(X) ≤ K_{1} log n. These estimates are sharp up to the values of the constants K and K_{1}. For a discussion see [20, pp. 86-92], [17], [16, II.14], or [38].

9 Local structure of infinite dimensional spaces

In this section we describe results and techniques whose purpose is to relate the structure of an infinite dimensional Banach space with the structure of its finite dimensional subspaces. We will be particularly interested in properties of a space which depend only on its family of finite dimensional subspaces and not on the way these finite dimensional spaces are “glued together” to form the infinite dimensional space.

A Banach space X is said to be λ-representable in a Banach space Y if for every finite dimensional subspace E of X and every ϵ > 0 there is a finite dimensional subspace F of Y with d(E, F) ≤ λ + ϵ. If this holds with λ = 1 we say that X is finitely representable in Y. In this terminology we can rephrase some results discussed in the previous section as follows: The space ℓ_{2} is finitely representable in every infinite dimensional Banach space Y (Dvoretzky’s theorem). A Banach space Y has cotype q for some q < ∞ if and only if c_{0} is not finitely representable in Y. A Banach space Y has type p for some p > 1 if and only if ℓ_{1} is not finitely representable in Y. It is a trivial fact that every Banach space is finitely representable in c_{0}.

For every Banach space X its second dual X^{∗∗} is finitely representable in X. In fact the following stronger statement, called the principle of local reflexivity, is true: Let E be a finite dimensional subspace of X^{∗∗}, F a finite dimensional subspace of X^{∗} and ϵ > 0. Then there is an operator T from E into T E ⊂ X so that ∥T ∥ ∥T^{−1}∥ ≤ 1 + ϵ, T_{|E∩X} is the identity and x^{∗∗}(x^{∗}) = x^{∗}(T x^{∗∗}) for every x^{∗} ∈ F and x^{∗∗} ∈ E.

Before giving a proof of the principle itself let us make some simple observations. If S : Y → Z is an operator with closed range then so is S + R if R is a finite rank operator. Also if S is as above and there is a y^{∗∗} ∈ Y^{∗∗} such that S^{∗∗}y^{∗∗} ∈ Z (i.e. the canonical image of Z in Z^{∗∗}) then there is for every δ > 0 a y ∈ Y such that ∥y∥ ≤ ∥y^{∗∗}∥(1 + δ) and S^{∗∗}y^{∗∗} = Sy. (This reduces immediately to the case where S is a quotient map and in this case it is obvious). Another simple observation is the following: for every ϵ there is a δ(ϵ) so that if (1 + δ)^{−1} ≤ ∥Sy∥ ≤ 1 + δ for every y in a δ-net in the unit sphere of Y, and some operator S, then (1 + ϵ)^{−1}∥y∥ ≤ ∥Sy∥ ≤ (1 + ϵ)∥y∥ for all y ∈ Y.

We pass to the proof of the principle of local reflexivity.

Let ϵ > 0 and let δ = δ(ϵ) be as above. Let E and F be finite dimensional subspaces of X^{∗∗}, respectively, X^{∗}. We pick {v_{j}^{∗}}_{j=1}^{m} in the unit ball of X^{∗} so that the set contains an algebraic basis of F and so that ∥x^{∗∗}∥ ≤ (1 + δ) max_{j} |x^{∗∗}(v_{j}^{∗})| for every x^{∗∗} ∈ E. Let {w_{i}^{∗∗}}_{i=1}^{n} be a δ-net in the unit sphere

of E so that {w_{i}^{∗∗}}_{i=1}^{k} is a basis of E ∩ X and {w_{i}^{∗∗}}_{i=1}^{r} is a basis of E for

some k ≤ r < n. Write w_{i}^{∗∗} = ∑_{h=1}^{r} λ_{i,h}w_{h}^{∗∗} for r < i ≤ n. For r < i ≤ n put

μ_{i,h} = λ_{i,h} if h ≤ r, μ_{ii} = −1, and μ_{i,h} = 0 for r < h ≠ i. Consider the

operator S : (\overbrace{X ⊕ ⋯ ⊕ X}^{n})_{∞} → (\overbrace{X ⊕ ⋯ ⊕ X}^{n−r+k})_{∞} defined by

S(x_{1}, …, x_{n}) = (x_{1}, …, x_{k}, ∑_{h=1}^{n} μ_{r+1,h}x_{h}, …, ∑_{h=1}^{n} μ_{n,h}x_{h}) .

By the choice of μ_{i,h} for h > r it follows that S is onto. Hence the operator

S̃ : (\overbrace{X ⊕ ⋯ ⊕ X}^{n})_{∞} → (\overbrace{X ⊕ ⋯ ⊕ X}^{n−r+h} ⊕ℝ^{nm})_{∞} defined by S̃(x_{1}, …, x_{n}) =

(S(x_{1}, …, x_{n}), v_{j}^{∗}(x_{h})), 1 ≤ j ≤ m, 1 ≤ h ≤ n has closed range. Note that by the definition of μ_{i,h} and the {w_{i}^{∗∗}}_{i=1}^{n},

S^{∗∗}(w_{1}^{∗∗}, w_{2}^{∗∗}, …, w_{n}^{∗∗}) = (w_{1}^{∗∗}, w_{2}^{∗∗}, …, w_{k}^{∗∗}, 0, 0, …) ∈ (\overbrace{X ⊕ ⋯ ⊕ X}^{n−r+k})_{∞}.

Hence by one of the observations above applied to the operator S̃ there exist {w_{i}}_{i=1}^{n} in X with sup_{i} ∥w_{i}∥ ≤ 1 + δ so that

S(w_{1}, w_{2}, …, w_{n}) = S^{∗∗}(w_{1}^{∗∗}, w_{2}^{∗∗}, …, w_{n}^{∗∗})

and w_{i}^{∗∗}(v_{j}^{∗}) = v_{j}^{∗}(w_{i}) for all i and j. Define now T : E → X by T w_{i}^{∗∗} = w_{i} for 1 ≤ i ≤ r. Note that T w_{i}^{∗∗} = w_{i} = w_{i}^{∗∗} for 1 ≤ i ≤ k. If r < i ≤ n then ∑_{h=1}^{r} μ_{i,h}w_{h} − w_{i} = the i’th component of S(w_{1}, w_{2}, …, w_{n}) = the i’th

component of S^{∗∗}(w_{1}^{∗∗}, w_{2}^{∗∗}, …, w_{n}^{∗∗}) = ∑_{h=1}^{r} μ_{i,h}w_{h}^{∗∗}−w_{i}^{∗∗} = 0. Hence T w_{i}^{∗∗} = w_{i}

also for r < i ≤ n. That T is an 1 + ϵ isometry follows from the choice of δ and the fact that for all i,

1 + δ ≥ ∥w_{i}∥ = ∥T w_{i}^{∗∗}∥ ≥ sup_{j} |v_{j}^{∗}(w_{i})| = sup_{j} |w_{i}^{∗∗}(v_{j}^{∗})| ≥ (1 + δ)^{−1}.

We describe next a useful construction in Banach space theory (having roots in mathematical logic) which is related to the notion of finite representability.

Recall that a family U of subsets of a set I is called a filter if it is closed under finite intersections, does not contain the empty set, and whenever A ⊂ B with A ∈ U then B ∈ U. A maximal (with respect to inclusion) filter is called an ultrafilter. By Zorn’s lemma every filter is contained in an ultrafilter. An ultrafilter is called free (or nontrivial) if the intersection of all sets in U is empty. An indexed family {x_{i}}_{i∈I} in a topological space is said to converge to x with respect to a filter U (in symbols, x = lim_{U} x_{i}) provided for every open set G containing x the set {i : x_{i} ∈ G} belongs to U. A Hausdorff space is compact if and only if every indexed family {x_{i}}_{i∈I} converges (to a unique point) for every free ultrafilter U on I. Assume now that I is a set and U is a free ultrafilter on I; assume also that for all i, X_{i} is a Banach space. We

define a seminorm |||·||| on (∑_{i} X_{i})_{∞} by |||x||| = lim_{U} ∥x_{i}∥ where x = {x_{i}}_{i∈I}

with x_{i} ∈ X_{i} for all i. The limit exists since a closed bounded interval on the line is compact. The quotient of (∑_{i} X_{i})_{∞} with respect to the closed subspace

of all x with |||x||| = 0 with its obvious norm is a Banach space, called the

ultraproduct of the X_{i} (with respect to U), and is denoted by (∏_{i} X_{i})_{U}. If all

the X_{i} are the same space X we call the space thus obtained an ultrapower of X, denoted also by X_{U}. Ultraproducts of Banach spaces are treated in detail in [9, Ch.8].

Given two families {X_{i}}_{i∈I} and {Y_{i}}_{i∈I} of spaces and operators T_{i} : X_{i} → Y_{i}

with sup_{i} ∥T_{i}∥ < ∞, there is a natural operator T : (∏_{i} X_{i})_{U} → (∏_{i} Y_{i})_{U} called the ultraproduct of the operators T_{i}. It maps an element in (∏_{i} X_{i})_{U}

represented by x = {x_{i}}_{i∈I} in (∑ X_{i})_{∞} into the element in (∏_{i} Y_{i})_{U} represented

by y = {T_{i}x_{i}}_{i∈I}.

The ultraproduct of one dimensional spaces is one dimensional and more generally if dim X_{i} = n < ∞ for all i then (∏_{i} X_{i})_{U} is also n-dimensional. On

the other hand if I = ℕ and lim_{U}(dim X_{i}) = ∞ then (∏_{i} X_{i})_{U} is already

nonseparable.

The ultraproduct of Banach lattices is again a Banach lattice if we take as the positive cone in (∏_{i} X_{i})_{U} the set of all elements which have representatives

x = {x_{i}}_{i∈I} in (∑_{i} X_{i})_{∞} with x_{i} ≥ 0 for all i. If all the X_{i} are abstract L_{p}

spaces for some fixed p, 1 ≤ p < ∞, then (∏_{i} X_{i})_{U} is again an abstract L_{p}

space and hence is isometric to L_{p}(μ) for some measure μ by the L_{p} version of the Kakutani representation theorem. Similarly, if all the X_{i} are C(K_{i})

spaces for some compact Hausdorff K_{i} then so is (∏_{i} X_{i})_{U}. However, for other

families of Banach spaces X_{i} (even e.g. if all are Orlicz spaces with the same

Orlicz function Φ) the determination of the nature of (∏_{i} X_{i})_{U} is not an easy

task.

As a first application of ultraproducts we shall prove now a fact mentioned already in Section 4: If 1 ≤ p ≤ r ≤ 2 then L_{r}(0, 1) is isometric to a subspace of L_{p}(0, 1). It was shown in Section 4 that for every integer n there is an isometry T_{n} : ℓ_{r}^{n} → L_{p}(0, 1). The ultraproduct T of the T_{i} is an isometry

from (∏_{n} ℓ_{r}^{n})_{U} into L_{p}(0, 1)_{U} where U is any free ultrafilter on the positive

integers. The space (∏_{n} ℓ_{r}^{n})_{U} contains L_{r}(0, 1) as a subspace. Indeed, by using

a sequence of partitions of [0,1] into n intervals (with each partition refining the previous one and so that the maximum length of the intervals tend to 0) we can represent L_{r}(0, 1) as \overline{⋃_{n} F_{n}} where F_{n} ⊂ F_{n+1} and F_{n} isometric to ℓ_{r}^{n}

for every n. For every g ∈ F_{n} let g̃ = (0, 0, …, 0, g, g, …, g …) ∈ (∑ F_{n})_{∞}

(n zeros). The closure of the images of all these g̃ in (∏_{n} F_{n})_{U} is isometric to

L_{r}(0, 1). Thus L_{r}(0, 1) is isometric to a subspace of the abstract L_{p}(μ) space L_{p}(0, 1)_{U}. The space L_{p}(μ) is nonseparable but, as in any L_{p}(μ) space, any separable subspace of it is isometric to a subspace of L_{p}(0, 1).

Every Banach space X is isometric in a natural way to a subspace of X_{U}; map x into the element represented by the element {x_{i}}_{i∈X} with x_{i} = x for all i. If X is a conjugate space (and in particular a reflexive space) there is a contractive projection from X_{U} onto the canonical image of X in it. Map {x_{i}}_{i∈I} to w^{∗}-lim_{U} x_{i}. As for duality, the space (∏_{i} X_{i}^{∗})_{U} can be in a natural

way identified with a subspace (usually proper) of the dual of (∏_{i} X_{i})_{U}.

We now discuss the relation between ultraproducts and λ-representability. It follows directly from the definitions that any ultrapower X_{U} of a Banach space X is finitely representable in X. There is also a converse, in a sense, to this statement. A Banach space X is λ-representable in Y if and only if there is a subspace Z of some ultraproduct of Y so that d(X, Z) ≤ λ. The only if part is clear from the remark above. The if part is an easy generalization of the remark made above (for L_{r}(0, 1)) that if X is contained in the closure of an increasing union of subspaces {X_{n}}_{n=1}^{∞} then X

is isometric to a subspace of (∏_{n} X_{n})_{U} for any free ultrafilter on the integers.

Indeed, let I be the set of pairs (E, ϵ) with E a finite dimensional subspace of X and ϵ > 0. Introduce a partial order on I by (E_{1}, ϵ_{1}) < (E_{2}, ϵ_{2}) if E_{1} ⊂ E_{2} and ϵ_{1} > ϵ_{2}, and let U be an ultrafilter on I containing for all i ∈ I the set {j ∈ I : i < j}. By assumption there is for every (E, ϵ) ∈ I an operator T_{E,ϵ} from E into Y so that ∥x∥ ≤ ∥T_{E,ϵ}x∥ ≤ (λ + ϵ)∥x∥ for all x ∈ E. For every

x ∈ X let x̃ = {x_{E,ϵ}} ∈ (∑_{i} Y )_{∞} be defined by x_{E,ϵ} = T_{E,ϵ}x if x ∈ E and

x_{E,ϵ} = 0 if x ∉ E. The image of all these x̃ in Y_{U} is easily seen to be a subspace Z which satisfies d(X, Z) ≤ λ.

From the principle of local reflexivity it follows that for every Banach space X the space X^{∗∗} is isometric to a norm one complemented subspace of a suitable ultraproduct of X. Without the complementation assertion this is a special case of the result above. To get the complementation assertion we have to modify the proof above. Now let I be the set of triples (E, F, ϵ), where E is a finite-dimensional subspace of X^{∗∗}, F a finite-dimensional subspace of X^{∗}, and ϵ > 0. Introduce a partial order on I by (E_{1}, F_{1}, ϵ_{1}) < (E_{2}, F_{2}, ϵ_{2}) if E_{1} ⊂ E_{2}, F_{1} ⊂ F_{2}, and ϵ_{1} > ϵ_{2}, and let U be an ultrafilter on I which refines the partial order filter. For every (E, F, ϵ) ∈ I let T_{E,F,ϵ} : E → X be the operator given by the principle of local reflexivity. Using these operators we define as above an isometry T from X^{∗∗} into X_{U}. Define a map S from X_{U} into X^{∗∗} by S({x_{i}}) = w^{∗}-lim_{U} x_{i}. From the properties of {T_{E,F,ϵ}} one deduces easily that ST is the identity on X^{∗∗}, so that T S is a projection of norm 1 from X_{U} onto T X^{∗∗}.

A property (P) of Banach spaces is called a super property provided that if X satisfies (P) and Y is finitely representable in X, then Y satisfies (P). In particular, a super property passes from a space X to all closed subspaces of its ultraproducts. So if (P) is a hereditary property (i.e., passes to closed subspaces), a Banach space X has super (P) if and only if every ultrapower of X has (P). For example, X is superreflexive if every ultrapower of X is reflexive. An explicit local property which characterizes superreflexivity is the following: A Banach space X is superreflexive if and only if for every ϵ > 0 there is an integer N(ϵ) so that any ϵ-separated dyadic tree in the unit ball of X has height ≤ N(ϵ). By an ϵ-separated dyadic tree of height N we mean a set of points {x_{i,n} : 1 ≤ i ≤ 2^{n}, 1 ≤ n ≤ N} so that x_{i,n} = (x_{2i−1,n+1} + x_{2i,n+1})/2 and ∥x_{2i−1,n+1} − x_{2i,n+1}∥ ≥ ϵ for every 1 ≤ i ≤ 2^{n} and every n < N. That this condition is necessary for superreflexivity is easy to see. A reflexive space cannot contain a bounded ϵ-separated dyadic tree {x_{i,n}} of height ∞ since the closed convex hull of such a tree is not dentable. The other direction of this implication follows from what we stated in section 6 concerning B-convexity. If X is nonreflexive it contains, for example, vectors {x_{j}}_{j=1}^{4} of norm 1 so that for

every i, ∥ ∑_{j=1}^{i} x_{j} − ∑_{j=i+1}^{4} x_{j}∥ ≥ 3. The set \frac{x_{1}+x_{2}+x_{3}+x_{4}}{4} , \frac{x_{1}+x_{2}}{2} , \frac{x_{3}+x_{4}}{2} , x_{1}, x_{2}, x_{3}, x_{4}

is a 1-separated dyadic tree of height three in the unit ball of X. In a similar manner we get in every nonreflexive space a 1-separated dyadic tree of an arbitrary finite height in the unit ball.

It is not hard to show that the existence of arbitrarily tall ϵ-separated trees in the unit ball (for some ϵ > 0 or for ϵ = 1) is a selfdual property and thus so is superreflexivity. Much deeper is the fact that a space is superreflexive if and only if it has an equivalent uniformly convex norm. Since uniform convexity is defined by an inequality involving four vectors the if part is obvious. The hard part is the only if part; that is, the construction of a uniformly convex equivalent norm in a space which does not have ϵ-separated dyadic trees of large height in its unit ball. The notion of dyadic trees reminds one of martingales and in fact the most elegant way to prove the only if part is to use vector-valued martingales. The proof shows in particular that if X is uniformly convex then there is an equivalent norm whose modulus of convexity δ(ϵ) satisfies δ(ϵ) ≥ Cϵ^{q} for some C > 0 and q < ∞. The proof of this fact and related material can be found in [2, 4.IV] and [6, IV.4] (see also [29]).

We introduce next a class of Banach spaces defined in terms of their finite dimensional subspaces which are closely related to L_{p}(μ) spaces. Let 1 ≤ p ≤ ∞ and λ ≥ 1. A Banach space X is called an L_{p,λ} space if for every finite dimensional subspace E of X there is a further subspace F ⊃ E with d(F, ℓ_{p}^{n}) ≤ λ where n = dim F. A space is called an L_{p} space if it is an L_{p,λ} space for some λ < ∞.

It is evident that every L_{p}(μ) space, 1 ≤ p ≤ ∞, is an L_{p,1+ϵ} space for every ϵ > 0 (take as F a small perturbation of a subspace spanned by a finite number of suitable disjoint indicator functions). Similarly, every C(K) space is an L_{∞,1+ϵ} space for every ϵ > 0 (use partitions of unity). By the principle of local reflexivity, if X^{∗∗} is an L_{p,λ} space then X is an L_{p,λ+ϵ} space for every ϵ > 0. From ultraproduct arguments used already above, it follows that: Any L_{p} space is isomorphic to a subspace of L_{p}(μ) for some measure μ (1 ≤ p ≤ ∞). In particular, for 1 < p < ∞ every L_{p} space is reflexive and any L_{2} space is isomorphic to a Hilbert space.

It is possible to make a stronger statement. Assume first that 1 < p < ∞ and that X is an L_{p,λ} space. There is thus a set I of finite dimensional subspaces E of X, directed by inclusion, with X = ⋃_{E∈I} E, so that for every E there are

operators T_{E} : E → ℓ_{p}^{n(E)}, S_{E} : ℓ_{p}^{n(E)} → E so that ∥T_{E}∥ ≤ 1, ∥S_{E}∥ ≤ λ and S_{E}T_{E} = I_{E} (n(E) = dim E). Let U be an ultrafilter on I which refines the order filter and put Y = (∏ ℓ_{p}^{n(E)})_{U}. Define T from X to Y by mapping x to the class represented by {T_{E}x}_{E∈I} (in view of the choice of U it does not matter that T_{E}x is defined only for E which contain x). Define S : Y → X by S{y_{E}} = w-lim_{U} S_{E}y_{E} (recall that X is reflexive). Then ST = I_{X} and Y is an L_{p}(μ) space. Consequently: Every L_{p} space X, 1 < p < ∞, is isomorphic to a complemented subspace of an L_{p}(μ) space. If X is separable we deduce that X is isomorphic to a complemented subspace of L_{p}(0, 1). Similar considerations (starting with the fact proved in section 4 that ℓ_{2} is isometric to a complemented subspace of L_{p}(0, 1)) yield that every Hilbert space is isometric to a complemented subspace of some L_{p}(μ) space when 1 < p < ∞.

In the cases p = 1 and p = ∞ we can reason similarly, but now define S̃ : Y → X^{∗∗} by S{y_{E}} = w^{∗}-lim_{U} S_{E}y_{E} and get that S̃T = J_{X}, the natural inclusion

of X into X^{∗∗}. By considering T^{∗∗} : X^{∗∗} → Y^{∗∗} and S̃^{∗∗} : Y^{∗∗} → X^{(iv)} and recalling that there is a norm one projection from X^{(iv)} onto X^{∗∗} (see section 2), it follows that if X is an L_{1} space then X^{∗∗} is isomorphic to a complemented subspace of an L_{1}(μ) space. Since L_{1}(μ)^{∗} is injective and since there is a norm one projection from X^{∗∗∗} onto X^{∗} we deduce that if X is an L_{1} space then X^{∗} is an injective Banach space. Similarly, if X is an L_{∞} space then the constructed ultraproduct Y is a C(K) space and X^{∗∗} is isomorphic to a complemented subspace of the injective space Y^{∗∗}. That is, if X is an L_{∞} space then X^{∗∗} is an injective Banach space.

The converse of the previous statements essentially hold. First we show: Let 1 < p < ∞. A Banach space X is isomorphic to a complemented subspace of an L_{p}(μ) space if and only if X is an L_{p} space or X is isomorphic to a Hilbert space. To see the “only if” direction, assume that there is a projection Q from Y = L_{p}(μ) onto a subspace X which is not isomorphic to a Hilbert space. By the dichotomy principle for L_{p} spaces, 2 < p < ∞, discussed in section 4 (and, by duality, also for 1 < p < 2), X has a subspace Z isomorphic to ℓ_{p} onto which there is a projection, say R. Let E be a finite dimensional subspace of X and ϵ > 0. There is a finite dimensional subspace F of Y (a small perturbation of the span of disjoint indicator functions) containing E so that d(F, ℓ_{p}^{m}) ≤ (1 + ϵ) (m = dim F) and so that there is a projection P from Y onto F with ∥P ∥ ≤ 1 + ϵ. The space RQF is a finite dimensional subspace of Z. Hence since every element in ℓ_{p} has essentially a finite support it follows that there is a subspace B of Z so that d(B, ℓ_{p}^{m}) ≤ K and max{∥x||, ∥y∥} ≤ K∥x + y∥ for every x ∈ QF and y ∈ B (K is a constant depending only on d(Z, ℓ_{p}); in fact, K ≤ 1 + ϵ is possible).

Let τ : F → B be an isomorphism from F onto B with ∥τ∥ ≤ 1 and ∥τ^{−1}∥ ≤ (1+ϵ)K. Define T : F → X by T = Q_{|F} +τ(I −P Q_{|F}). Then the restriction of T to E is the identity (since this is true of the restrictions of P and Q to E). A simple computation shows that for every x ∈ F, (1 + ϵ)^{−1}(1 + K)^{−1}K^{−1}∥x∥ ≤ ∥T x∥ ≤ (1 + (2 + ϵ)∥Q∥)∥x∥. Hence G = T F is a subspace containing E with d(G, ℓ_{p}^{m}) ≤ (1 + ϵ)(1 + K)K(1 + (2 + ϵ)∥Q∥).

Consider now the case p = 1. Every infinite dimensional complemented subspace of an L_{1}(μ) contains a subspace isomorphic to ℓ_{1}. This is a consequence of the fact that such a space is nonreflexive (to be proved in the next section) and the structure of nonweakly compact sets in L_{1}(μ) discussed in section 4. Hence if X^{∗} is injective then X^{∗∗}, being a complemented subspace of L_{1}(μ) for some μ, contains a copy of ℓ_{1}. The proof given above for 1 < p carries over verbatim and yields that X^{∗∗} and hence X is an L_{1} space.

It is also true (see [21, II.D]) that an infinite dimensional complemented subspace of a C(K) space (and hence every infinite dimensional injective space) contains a copy of c_{0}. The same proof then yields that every space whose second dual is injective is an L_{∞} space. Instead of using this moderately difficult structure theorem for C(K) one can argue that if X^{∗∗} is injective then X^{∗∗∗} is complemented in some L_{1}(μ) space, hence X^{∗∗∗} contains a complemented subspace isomorphic to ℓ_{1}, whence X^{(iv)} contains a subspace isomorphic to c_{0}.

One consequence of the principle of local reflexivity (see [24]) is: If X^{∗} has the BAP then X has the BAP. Since every L_{p}(μ) space, 1 ≤ p ≤ ∞, has the BAP and the BAP passes to complemented subspaces, we conclude: Every L_{p} space has the BAP. A stronger statement is mentioned at the end of this section.

From the results above it follows immediately that any complemented subspace of an L_{p} space 1 ≤ p ≤ ∞ is again an L_{p} space unless it is isomorphic to a Hilbert space (for p = 1 or p = ∞ it cannot be a Hilbert space). We also get that: the dual of an L_{p} space is an L_{p^{∗}} space. This fact is not evident from the definition since L_{p} spaces are defined by the structure of their finite dimensional subspaces and by passing to the duals we get direct information only on the finite dimensional quotient spaces.

The L_{p} spaces give a nice local description of the complemented subspaces of L_{p}(0, 1) for 1 < p < ∞. From the global point of view these complemented subspaces are very hard to describe. Some global structure theorems are known (e.g. every separable L_{p} space 1 ≤ p ≤ ∞ has a Schauder basis, and every separable infinite dimensional L_{p} space 1 < p < ∞ which does not contain a copy of ℓ_{2} is isomorphic to ℓ_{p}) but many natural questions on them are still open. For example, it is unknown whether every separable L_{p} space, 1 < p < ∞, (p ≠ 2), has an unconditional basis. There are uncountably many distinct isomorphism types among the separable L_{p} spaces, 1 ≤ p ≤ ∞, (p ≠ 2). The simplest examples for 1 < p ≠ 2 < ∞ (besides L_{p}(μ) spaces) are ℓ_{p} ⊕ ℓ_{2} and (ℓ_{2} ⊕ ℓ_{2} ⊕ ⋯)_{p}. A discussion of all of these questions is contained in [22].

Among the separable L_{∞} spaces there are the C(K) spaces with K countable compact metric which, as was pointed out in section 4, form uncountably many distinct isomorphism types. There are however spaces X such that X^{∗} is isometric to ℓ_{1} but X is not isomorphic to a C(K) space. Such a space X is an L_{∞,1+ϵ} space for every ϵ > 0. In fact, a space X is L_{∞,1+ϵ} for every ϵ > 0 if and only if X^{∗} is isometric to L_{1}(μ) for some measure μ. It is known that such an X has a subspace isometric to c_{0}. In contradistinction to this, there are L_{∞,λ} spaces with λ > 1 which have the RNP (and thus in particular do not contain a subspace isomorphic to c_{0}). For a discussion of L_{∞} spaces see [40].

There are a continuum of isomorphism classes of separable L_{1} spaces. The simplest example (besides ℓ_{1} and L_{1}(0, 1)) is the kernel of a quotient map T : ℓ_{1} → L_{1}. The isomorphism type of this kernel turns out not to depend on the choice of T. This kernel is an L_{1} space which is not isomorphic to a complemented subspace of an L_{1}(μ) space. For a discussion of L_{1} spaces, see ([22]).

There is also a “local view” of Banach lattices. A Banach space X is said to have Gordon-Lewis local unconditional structure (GL-l.u.st.) if there is a constant λ so that for every finite dimensional subspace E of X there is a space Y with an unconditional basis and operators: T : E → Y, S : Y → X so that ST = I_{E} and ∥T ∥ ∥S∥uc(Y) ≤ λ where uc(Y) denotes the unconditional constant of a basis in Y. Note that by a simple perturbation argument we can assume that Y is always finite dimensional (we may have to replace λ by any λ′ > λ). If always Y can be chosen to be a subspace of X which contains E, X is said to have local unconditional structure (l.u.st.). Then of course T and S can be taken to be the inclusion maps. It is evident that l.u.st. implies GL-l.u.st., but whether the converse is true is open. Later in this section we mention a partial converse.

Any Banach lattice X has l.u.st. where the λ in the definition can be taken as any constant > 1. Indeed, if X is an order complete lattice, then we saw in section 5 that for any finite dimensional subspace E and every ϵ > 0 there are vectors {x_{1}, …, x_{m}} in X whose span contains E and which is an ϵperturbation of a sequence of disjointly supported vectors. Since a disjointly supported sequence in a lattice is 1-unconditional and the bidual of a Banach lattice is an order complete lattice, the general case follows from the principle of small perturbations and the principle of local reflexivity.

It is clear from the definition that L_{p} spaces, 1 ≤ p ≤ ∞, have l.u.st.

The close relationship between l.u.st. and lattice structure is expressed in the following result: A Banach space X has GL-l.u.st. if and only if X^{∗∗} is isomorphic to a complemented subspace of a Banach lattice. Indeed, since by the definition a complemented subspace of a space with GL-l.u.st. has GL-l.u.st., it follows that if X^{∗∗} is a complemented subspace of a lattice it has GL-l.u.st. By the principle of local reflexivity we deduce that also X has GL-l.u.st.

To prove the other direction we use ultraproducts. Assume X has GL-l.u.st. with some constant λ. For every finite dimensional subspace E of X there are a space Y_{E} with 1-unconditional basis and operators T_{E} : E → Y_{E} and S_{E} : Y_{E} → X with ∥T_{E}∥ ≤ 1, ∥S_{E}∥ ≤ λ and S_{E}T_{E} = I_{E}. Let I be the directed set of all finite dimensional subspaces of X and let U be an ultrafilter on I which refines the order filter. Let T : X → Y = (∏ Y_{E})_{U} be defined by T x = {T_{E}x} (as before, because of our assumption on U it does not matter that T_{E}x is defined just for the E containing x). Let S : Y → X^{∗∗} be defined by S{y_{E}} = w^{∗} − lim S_{E}y_{E}. The space Y is a Banach lattice and ST = I_{X}. Hence T^{∗∗} is an isomorphism from X^{∗∗} into the lattice Y^{∗∗} and P S^{∗∗} is a projection from Y^{∗∗} onto T^{∗∗}(X^{∗∗}) where P is a projection from X^{(iv)} onto X^{∗∗} (see section 2).

An immediate consequence of the claim just proved is that X has GL-l.u.st. if and only if X^{∗} has GL-l.u.st. Indeed, if X has GL-l.u.st. then X^{∗∗} is complemented in a Banach lattice. Hence the same is true for X^{∗∗∗} and thus X^{∗} has GL-l.u.st. Conversely, if X^{∗} has GL-l.u.st. then X^{∗∗∗}, hence also X^{(iv)}, is complemented in a Banach lattice. Since X^{∗∗} is complemented in X^{(iv)}, it follows that also X has GL-l.u.st.

The argument given earlier that if a complemented subspace X of an L_{p}(μ) space is not isomorphic to a Hilbert space then it is an L_{p} space depended on the fact that an L_{p}(μ) space is finitely representable in ℓ_{p} and that ℓ_{p} is, for some λ, λ-finitely representable in every finite codimensional subspace of X. The same argument shows that if Y is complemented in a Banach lattice X and there is λ < ∞ so that X is λ-finitely representable in every finite codimensional subspace of Y, then Y has l.u.st. From this and the characterization of GL-l.u.st. mentioned above we deduce: X has GL-l.u.st. if and only if X ⊕ c_{0} has l.u.st.

Among the spaces which fail to have GL-l.u.st. important examples are the spaces of compact and bounded operators on ℓ_{2}. In fact, the only unitary ideal of operators on ℓ_{2} (a notion discussed in section 10) which has GL-l.u.st. is the space of Hilbert-Schmidt operators, which is itself a Hilbert space. Spaces of analytic functions, such as the disc algebra, typically do not have GL-l.u.st. For a proof of most of these statements see [9, Ch.17], [20], or [27].

We conclude this section by mentioning briefly a quantitative notion of the approximation property and its relation to ultrapowers. A Banach space X is said to have the uniform approximation property (UAP) if there is a constant λ and a uniformity function φ(·) on the natural numbers so that for every finite dimensional subspace E of X there is T ∈ B(X, X) so that ∥T ∥ ≤ λ, T_{|E} = I_{E}, and dim T X ≤ φ(dim E). A Banach space X has the UAP if and only if every ultrapower of X has the BAP. It follows from this that every L_{p} space, 1 ≤ p ≤ ∞, has the UAP. It also follows, in view of results about ultrapowers presented earlier, that if X^{∗∗} fails the BAP then X fails the UAP. From this it can be deduced (see [24]) that there are Banach spaces which have the BAP but not the UAP (in fact, K(ℓ_{2}, ℓ_{2}) is such a space).

While it is usually easy to check whether a concrete separable space has the BAP, it is often quite difficult to decide whether a special space has the UAP. For example, it is open whether the disk algebra has the UAP. As a matter of fact, the only general class of spaces other than L_{p} spaces which are known to have the UAP are the reflexive Orlicz spaces.

10 Some special classes of operators

In this section we shall discuss some special classes of linear operators between Banach spaces and their relation to the geometry of Banach spaces. We shall also discuss some other topics in operator theory which are relevant to the study of the structure of Banach spaces.

Most of the classes of operators we consider have the ideal property, meaning that the operators from a fixed X into a fixed Y having this property form a linear space and that whenever T : X → Y belongs to this class then for every bounded U : Z → X and V : Y → W the operator V T U belongs to this class. The three most elementary operator ideals are the bounded operators, the compact operators, and the weakly compact operators. Since B(X, Y), K(X, Y), and W K(X, Y) are all Banach spaces under the operator norm, the operator norm is the natural norm to use for operators in these classes. A Banach space X has the Dunford-Pettis (DP) property provided that every weakly compact operator with domain X maps weakly compact sets into norm compact sets. It is clear that if T ∈ W K(X, Y), S ∈ W K(Y, Z), and Y has the DP property, then ST is a compact operator. In particular, if X has the DP property and P is a weakly compact projection on X, then P = P^{2} is compact. This means that the only complemented reflexive subspaces of X are the finite dimensional ones.

A subset of a Banach space is relatively weakly compact if and only if it is relatively weakly sequentially compact. Therefore, a space X has the DP property if and only if every weakly compact operator with domain X maps sequences which converge weakly to zero into sequences which converge in norm to zero.

The following is an elegant characterization of spaces having the DP property. X has the DP property if and only if whenever {x_{n}}_{n=1}^{∞} in X tends weakly to 0 and {x_{n}^{∗}}_{n=1}^{∞} in X^{∗} tends weakly to 0 the sequence of scalars {x_{n}^{∗}(x_{n})}_{n=1}^{∞} tends to 0. Indeed, assume that X has DP and x_{n}^{∗} ⟶^{w} 0. The operator T : X → c_{0} defined by T x = (x_{1}^{∗}(x), x_{2}^{∗}(x), …) is a weakly compact operator (it is easier to check that T^{∗} is weakly compact). Hence, if x_{n} → 0 weakly, |x_{n}^{∗}(x_{n})| ≤ ∥T x_{n}∥ → 0. Conversely, assume the condition on the sequences is satisfied, T : X → Y is weakly compact, and x_{n} ⟶^{w} 0. If ∥T x_{n}∥ ↛ 0 we may assume, by passing to a subsequence, that ∥T x_{n}∥ ≥ δ for some δ > 0 and all n. Let y_{n}^{∗} ∈ X^{∗} be such that y_{n}^{∗}(T x_{n}) = ∥T x_{n}∥ and ∥y_{n}^{∗}∥ = 1 for all n. Since T^{∗} is weakly compact as well we may assume (passing again to a subsequence if needed) that T^{∗}y_{n}^{∗} ⟶^{w} x^{∗} for some x^{∗}. Then

0 = lim_{n}(T^{∗}y_{n}^{∗} − x^{∗})(x_{n}) = lim_{n} y_{n}^{∗}(T x_{n}) = lim_{n} ∥T x_{n}∥

a contradiction.

It follows from the criterion we just proved that X has the DP property if X^{∗} has this property. (The converse is false: ( ∑_{n=1}^{∞} ℓ_{2}^{n})_{1} has the DP property

but ( ∑_{n=1}^{∞} ℓ_{2}^{n})_{∞} contains a complemented copy of ℓ_{2} and hence fails the DP

property.) It is also clear that a complemented subspace of a space with the DP property has the same property. We shall verify shortly that C(K) spaces have the DP property. Consequently we get that: All L_{∞} and L_{1} spaces have the DP property.

To show that C(K) has the DP property let {x_{n}}_{n=1}^{∞} be a sequence in the unit ball of C(K) so that x_{n}(t) → 0 for every t ∈ K. Let {μ_{n}}_{n=1}^{∞} be a sequence in the unit ball of C(K)^{∗} which tends weakly to 0. All the μ_{n} can be considered as elements in L_{1}(μ) where μ = ∑_{n=1}^{∞} |μ_{n}|/2^{n}. By the analysis done in section 4

of sets in L_{1}(μ) which have weakly compact closure it follows that for every ϵ > 0 there is a δ > 0 so that whenever E is a Borel set in K with μ(E) < δ then |μ_{n}|(E) < ϵ for all n. By Egoroff’s theorem there is a set E with μ(E) < δ so that |x_{n}(t)| ≤ ϵ for all t outside E and n ≥ n(ϵ). Hence for such n,

|∫_{K} x_{n}dμ_{n}| ≤ |∫_{E} x_{n}dμ_{n}| + | ∫_{K∼E} x_{n}dμ_{n}| ≤ ϵ + ϵ;

that is, ∫_{K} x_{n}dμ_{n} → 0.

We also want to single out the strictly singular operators. An operator T : X → Y is strictly singular if it is not an isomorphism on any infinite dimensional subspace of X. Of course, any compact operator is strictly singular but the converse is false. The formal identity map from ℓ_{p} into ℓ_{r} where p < r is a simple example of a strictly singular operator which is not compact. Note that, by the discussion in section 3, an operator from ℓ_{p}, 1 ≤ p < ∞, into itself (or from c_{0} into itself) is strictly singular if and only if it is compact. A more sophisticated and interesting example of a strictly singular operator is the formal identity operator I_{p} : C(K) → L_{p}(μ), 1 ≤ p < ∞, where μ is a finite measure on the compact set K. The operator I_{p} is not compact if, for example, K = [0, 1] and μ is Lebesgue measure (in fact, I_{p} is compact only if μ is purely atomic). Observe that if X is a subspace of C(K), the statement that the restriction of I_{p} to X is an isomorphism means that the supremum norm and the L_{p}(μ) norm are equivalent on X. If this occurs for some p < ∞, then the extrapolation argument used in section 4 shows that the supremum norm and the L_{r}(μ) norm are equivalent on X for all 1 ≤ r < ∞. Thus it is enough to check e.g. that I_{1} or I_{2} is strictly singular. That I_{2} is strictly singular is an exercise in textbooks (e.g. [18, Ch. 10 # 41, # 55]), but we prove it later in this section when we discuss p-summing operators.

Unlike the situation with compact or weakly compact operators, the dual of a strictly singular operator need not be strictly singular. If T is a quotient map from ℓ_{1} onto a separable space X not containing a subspace isomorphic to ℓ_{1} (e.g. X = ℓ_{2}) then T is strictly singular but T^{∗} is an isometric embedding of X^{∗} into ℓ_{∞}.

Although strictly singular operators need not be compact they have compact restrictions to infinite dimensional subspaces. Indeed, by the same method used in section 3 to prove the existence of a basic sequence in any Banach space one can prove the following. If T : X → Y is not an isomorphism on any subspace of X of finite codimension then there is for every ϵ > 0 a normalized basic sequence {x_{n}}_{n=1}^{∞} in X (with basis constant two, say) so that ∥T x_{n}∥ ≤ 4^{−n}ϵ for every n ≥ 1. The restriction of T to the span of {x_{n}}_{n=1}^{∞} is a compact operator of norm ≤ ϵ.

It follows from this observation that the sum of two strictly singular operators is strictly singular and thus the class of strictly singular operators has the ideal property. It is also easy to check that SS(X, Y) is complete in the operator norm.

Two Banach spaces X and Y are called totally incomparable if there is no infinite dimensional space Z which is isomorphic to subspaces of both X and Y. If every operator from X to Y is strictly singular (and in particular if X and Y are totally incomparable) and Z is a complemented subspace of X ⊕ Y then there is an automorphism T of X ⊕ Y so that T Z is of the form T Z = X_{0} ⊕ Y_{0} where X_{0} (respectively Y_{0}) is a complemented subspace of X (respectively Y). This result is not as simple as the preceding results in this section. Its proof can be found in [11] 2.c.13.

Another class of operators is that of Fredholm operators. An operator T : X → Y is called a Fredholm operator if α(T) := dim ker T < ∞ and T X is closed with β(T) := dim Y/T X < ∞. The number i(T) = α(T) − β(T) is called the index of T and has a significant role in many applications. As a matter of fact i(T) can be defined also for non-Fredholm operators provided T X is closed and at least one of the numbers α(T) or β(T) is finite (in this case the index can be of course either +∞ or −∞). The class of Fredholm operators is not closed under addition or composition. However if T_{1} and T_{2} are Fredholm operators so is T_{1}T_{2} (provided it is properly defined) and i(T_{1}T_{2}) = i(T_{1}) + i(T_{2}). Also i(T^{∗}) = −i(T). These facts are simple exercises in linear algebra. Somewhat harder to prove is that F r(X, Y) is preserved by strictly singular perturbations. If T, S : X → Y with T Fredholm and S strictly singular then T + S is also Fredholm with i(T + S) = i(T) (see [11] 2.c.9).

An operator T : X → Y is called absolutely summing if whenever ∑_{i=1}^{∞} x_{i}

converges unconditionally in X the series ∑_{i=1}^{∞} T x_{i} converges absolutely. From

the closed graph theorem one deduces easily that T is absolutely summing if and only if there is a constant C so that for any choice of {x_{i}}_{i=1}^{n} in X

∑_{i=1}^{n} ∥T x_{i}∥ ≤ C sup {∑_{i=1}^{n} |x^{∗}(x_{i})| : ∥x^{∗}∥ ≤ 1} .

More generally, for every 0 < p < ∞ we can define the class of p-summing operators as those operators for which there is a constant C so that for all choices of {x_{i}}_{i=1}^{n} in X

(∑_{i=1}^{n} ∥T x_{i}∥^{p})^{1/p} ≤ C sup {(∑_{i=1}^{n} |x^{∗}(x_{i})|^{p})^{1/p} : ∥x^{∗}∥ ≤ 1} . (28)

Thus 1-summing is the same as absolutely summing. The smallest constant C for which (28) holds is denoted by π_{p}(T). If T is not p-summing we put π_{p}(T) = ∞. The set of all T : X → Y with π_{p}(T) < ∞ is denoted by Π_{p}(X, Y). It is trivial to check that for 1 ≤ p < ∞, Π_{p}(X, Y) is a Banach space with π_{p}(T) as the norm. It is also clear from the definition that whenever ST is defined and one of the operators S, T is p-summing so is their product and π_{p}(ST) ≤ ∥S∥π_{p}(T) and π_{p}(ST) ≤ π_{p}(S)∥T ∥. In particular the class of p summing operators has the ideal property. Sometimes one uses also the notation π_{∞}(T). It naturally means just the usual operator norm ∥T ∥ of T.

Without further mention we shall henceforth assume that p ≥ 1 when discussing p-summing operators. However, there are important applications to Banach space theory of p-summing operators for 0 < p < 1 (see, e.g., the article [32]), and much of the theory of p-summing operators, 1 ≤ p < ∞, goes over to the range 0 < p < 1. In particular, there is a version of the Pietsch factorization theorem (discussed below when 1 ≤ p < ∞) for p-summing operators, 0 < p < 1 (see [20, Th. 9.1]).

Notice that the supremum on the right side of (28) is the norm of the operator from ℓ_{p^{∗}}^{n} to X defined by e_{k} ↦ x_{k} (notice that p ≥ 1 is needed for this). Thus π_{p}(T) can also be defined by

π_{p}(T) = sup {(∑_{n=1}^{∞} ∥T V e_{n}∥^{p})^{1/p} : V : ℓ_{p^{∗}} → X, ∥V ∥ ≤ 1} . (29)

Equation (29) implies that π_{p}(T) is the supremum of π_{p}(T V) as V varies over the norm one operators from ℓ_{p^{∗}} into X.

In contrast to the classes of weakly compact or strictly singular operators, the class of p-summing operators is determined by the behavior of T on the finite dimensional subspaces of X. Because of the quantitative nature of its definition the notion of p-summing norm plays also an important role in the study of finite dimensional spaces.

If K is a compact Hausdorff space and μ a regular probability measure on K then the identity operator I_{p} from C(K) into L_{p}(μ) is easily seen to be a p-summing operator with p-summing norm one. It turns out that this simple example is the prototype of general p-summing operators. This is the content of the Pietsch factorization theorem. Let X be a subspace of C(K). An operator T : X → Y is p-summing, 1 ≤ p < ∞, if and only if there is a regular probability measure μ on K and a constant C so that for all x in X,

||T x∥ ≤ C ( ∫_{K} |x(t)|^{p}dμ(t))^{1/p} . (30)

Moreover, the smallest C for which (30) holds is π_{p}(T). Note also that the “only if” part can always be applied with K = B_{X^{∗}} under the weak^{∗} topology, or with K the weak^{∗} closure of the extreme points of B_{X^{∗}}, again in the weak^{∗} topology.

In diagrammatic form, the Pietsch factorization can be stated as follows. Let X be a subspace of C(K). If T is p-summing then there is a regular probability measure μ on K and an operator S from X_{p} := \overline{I_{p}X} into Y so that T = SI_{p|_{X}}; moreover, ∥S∥ = π_{p}(T).

C(K) ⟶^{I_{p}} L_{p}(μ)
∪  ∪
X ⟶^{I_{p|X}} X_{p}
↘^{T} ↓ S
Y

(For p = 2, the operator S can be extended to an operator from L_{p}(μ) into Y, but for p ≠ 2 this is not in general the case.)

To prove the Pietsch factorization theorem, assume that π_{p}(T) = 1. Consider the following subsets of C(K)

G = {f ∈ C(K) : sup{f(t) : t ∈ K} < 1}
F = conv{f ∈ C(K) : f(·) = |x(·)|^{p}, ∥T x∥ = 1}.

The sets F and G are convex with G open and, since π_{p}(T) = 1, F ∩ G = ∅. By the separation theorem there is a regular signed measure μ on K and a positive λ so that ∫ f dμ < λ for f ∈ G and ∫ f dμ ≥ λ for f ∈ F. Since whenever g ≤ f and f ∈ G also g ∈ G it follows that the measure μ has to be positive so after normalization we may assume that it is a probability measure. Also since G contains the open unit ball of C(K) we must have λ ≥ 1. Hence

for every x ∈ X, ∥T x∥^{p} ≤ ∫_{K} |x(t)|^{p}dμ(t).

From this factorization theorem we can easily deduce several interesting facts. First, whenever 1 ≤ p ≤ r < ∞, π_{r}(T) ≤ π_{p}(T) and thus Π_{p}(X, Y) ⊂ Π_{r}(X, Y). (This is trivially true also for r = ∞.) Next, since L_{p}(μ) is reflexive for 1 < p < ∞, every operator in Π_{p}(X, Y) is weakly compact. By the preceding remark this is true also for p = 1. A third consequence is that the identity operator on an infinite dimensional space X is never p-summing for any p < ∞. Indeed, by the Pietsch factorization theorem, this is equivalent to the assertion that I_{p} : C(K) → L_{p}(μ) is strictly singular for any probability measure μ on any compact Hausdorff space K. As we mentioned earlier, if I_{p} were not strictly singular for some p < ∞, then there would exist an infinite dimensional subspace X of C(K) so that for all 1 ≤ p < ∞, the L_{p}(μ) norm is equivalent to the supremum norm. So X would be isomorphic to an infinite dimensional Hilbert space and the identity operator on X (hence also the identity operator on ℓ_{2}) would be 1-summing. But obviously there exists in ℓ_{2} an unconditionally converging series which is not absolutely summable.

Recall that we already proved in section 8 that the identity operator on an infinite dimensional space is not absolutely summing (in fact, a stronger statement was proved). The two proofs we have given for this result are very different but have in common that Hilbert spaces enter naturally into them.

There are many other places where the Pietsch factorization theorem is useful, in particular, for proving the composition inequality for p-summing operators, which says if T ∈ Π_{p}(X, Y) and S ∈ Π_{q}(Y, Z), then ST ∈ Π_{r}(X, Z) and π_{r}(ST) ≤ π_{p}(T)π_{q}(S), where 1/r := 1 ∧ (1/p + 1/q). See ([9, 2.22]). The Pietsch factorization theorem also makes it easy to check that π_{p}(T^{∗∗}) ≤ π_{p}(T) for any operator T (see [9, 2.19]). This fact can also be proved using the principle of local reflexivity.

Next we examine ideals of operators on ℓ_{2} and how they are related to psumming operators. If T is a compact operator on a Banach space then we can define a (finite or infinite) sequence {λ_{n}(T)}_{n=1}^{∞} consisting of the nonzero eigenvalues of T, repeated according to multiplicity, and ordered so that |λ_{1}(T)| ≥ |λ_{2}(T)| ≥ · · ·. If the sequence is infinite, then necessarily λ_{n}(T) → 0. For the Banach space ℓ_{2}, the spectral theorem for compact operators says that if T ∈ K(H, H), then there is an orthonormal basis {x_{n}}_{n=1}^{∞} so that the selfadjoint operator T^{∗}T is represented as

T^{∗}T x = ∑_{n∈P} λ_{n}(T^{∗}T)⟨x, x_{n}⟩x_{n} (31)

with λ_{n}(T^{∗}T) > 0 for n ∈ P and T^{∗}T x_{n} = 0 for n ∉ P.

Now if X is a Banach space with a 1-symmetric basis S = {e_{n}}_{n=1}^{∞}, let S(ℓ_{2}) be those compact operators on ℓ_{2} for which ∑ \sqrt{λ_{n}(T^{∗}T)} e_{n} converges in X, and, for T ∈ S(ℓ_{2}), set

σ_{S}(T) := ∥∑ \sqrt{λ_{n}(T^{∗}T)}e_{n}∥_{X} . (32)

Then (S(ℓ_{2}), σ_{S}) is a Banach space (verifying the triangle inequality is a bit tricky). It is less difficult to check that S(ℓ_{2}) satisfies the unitary ideal property

σ_{S}(UT V) ≤ ∥U∥σ_{S}(T)∥V ∥

for all U, V ∈ B(H, H) and T ∈ S(ℓ_{2}).

When S is the unit vector basis for ℓ_{p}, 1 ≤ p < ∞, the resulting ideal, called the Schatten-von Neumann p-class, is denoted by (S_{p}, σ_{p}). The ideal (S_{2}, σ_{2}), which is a Hilbert space, is called the Hilbert-Schmidt class. The Schattenvon Neumann classes are discussed in many books, including [9] and [21].

Given the representation (31) for T^{∗}T with T ∈ K(H, H), it is evident that {T x_{n}}_{n=1}^{∞} is orthogonal and ∥T x_{n}∥^{2} = λ_{n}(T^{∗}T), and hence σ_{p}(T) = (∑_{n=1}^{∞} ∥T x_{n}∥^{p})^{1/p} for 1 ≤ p < ∞. In the Hilbert-Schmidt case, σ_{2}(T) = (∑_{n=1}^{∞} ∥T y_{n}∥^{2})^{1/2} for any orthonormal basis {y_{n}}_{n=1}^{∞} (just write ∥T y_{m}∥^{2} = ⟨T^{∗}T y_{m}, y_{m}⟩ and expand y_{m} with respect to {x_{n}}_{n=1}^{∞}). From this and the ideal property of S_{2} it follows easily that Π_{2}(ℓ_{2}, ℓ_{2}) = S_{2} with equality of the respective norms. Indeed, if T ∈ K(ℓ_{2}) and {x_{n}}_{n=1}^{∞} is an orthonormal basis, then by (29) σ_{2}(T)^{2} = ∑_{n=1}^{∞} ∥T x_{n}∥^{2} ≤ π_{2}(T)^{2}. On the other hand, if V : ℓ_{2} → ℓ_{2}

and ∥V ∥ ≤ 1, then (∑_{n=1}^{∞} ∥T V x_{n}∥^{2})^{1/2} ≤ σ_{2}(T V) ≤ σ_{2}(T), so (29) gives π_{2}(T) ≤ σ_{2}(T).

For an operator T : ℓ_{2} → X, it is not true that π_{2}(T)^{2} = ∑ ∥T x_{n}∥^{2} for every orthonormal basis {x_{n}}_{n=1}^{∞}, but it is not very hard to show that π_{2}(T)^{2} is the supremum over all orthonormal bases of ∑ ∥T x_{n}∥^{2}. More importantly, there is a finite version [20, 18.4] of this which is useful for studying finite dimensional spaces. This lemma says: If T is an operator from ℓ_{2}^{n} into some Banach space, then there is an orthonormal basis {x_{i}}_{i=1}^{n} so

that π_{2}(T)^{2} ≤ 2 ∑_{i=1}^{n} ∥T x_{i}∥^{2}.

We next discuss p-summing operators on ℓ_{2} for other values of p: For every 1 ≤ p < ∞, the p-summing operators on ℓ_{2} coincide with the Hilbert-Schmidt operators, and of course π_{p}(·) is equivalent to σ_{2}(·). First a preliminary remark. Given T ∈ K(H, H) with T^{∗}T represented as in (31), define |T |x = ∑ \sqrt{λ_{n}(T^{∗}T)}⟨x, x_{n}⟩x_{n}. There is an obvious partial isometry (i.e., an orthogonal projection followed by an isometry) U so that T = U|T |. This is the polar decomposition of the compact operator T. Suppose now that T is Hilbert-Schmidt. We want to see that T is 1-summing (and hence p-summing for every 1 ≤ p < ∞). By considering the polar decomposition of T, we can assume that there is an orthonormal basis {e_{n}}_{n=1}^{∞} and λ_{n} ≥ 0 so that T e_{n} = λ_{n}e_{n} for every n and 1 = σ_{2}(T) = ∑_{n=1}^{∞} λ_{n}^{2}. Let {ϵ_{n}}_{n=1}^{∞} be a Rademacher

sequence. Using Khintchine’s inequality (1) we get

sup {∑_{i=1}^{n} |⟨x_{i}, x⟩| : ∥x∥ ≤ 1} ≥ sup_{±} {∑_{i=1}^{n} |⟨x_{i}, ∑_{j=1}^{∞} ±λ_{j}e_{j}⟩|}

≥ ∑_{i=1}^{n} 𝔼 |∑_{j=1}^{∞} λ_{j}ϵ_{j}⟨x_{i}, e_{j}⟩|

≥ A_{1} ∑_{i=1}^{n} (∑_{j=1}^{∞} |λ_{j}⟨x_{i}, e_{j}⟩|^{2})^{1/2} = A_{1} ∑_{i=1}^{n} ∥T x_{i}∥.

This shows that π_{1}(T) ≤ A_{1}^{−1}σ_{2}(T).

To show the converse, it is enough to check that if T ∈ Π_{p}(ℓ_{2}, ℓ_{2}) with 2 ≤ p < ∞ then T is 2-summing. More generally: If T ∈ Π_{p}(X, Y) with 2 ≤ p < ∞ and Y has cotype 2 then T ∈ Π_{2}(X, Y). Since T is p-summing, the Pietsch factorization theorem kindly supplies a probability measure μ on B_{X^{∗}} so that for all x in X,

∥T x∥ ≤ π_{p}(T) ( ∫_{B_{X^{∗}}} |x^{∗}(x)|^{p}dμ(x^{∗}))^{1/p} .

Using Khintchine’s inequality (1) in the last step, we get

(∑_{i=1}^{n} ∥T x_{i}∥^{2})^{1/2} ≤ C_{2}(Y) (𝔼 ∥∑_{i=1}^{n} ϵ_{i}T x_{i}∥^{2})^{1/2}

≤ C_{2}(Y) (𝔼 ∥T (∑_{i=1}^{n} ϵ_{i}x_{i})∥^{p})^{1/p}

≤ C_{2}(Y)π_{p}(T) (𝔼 ∫_{B_{X^{∗}}} |x^{∗} (∑_{i=1}^{n} ϵ_{i}x_{i})|^{p} dμ(x^{∗}))^{1/p}

≤ C_{2}(Y)π_{p}(T)B_{p} sup {(∑_{i=1}^{n} |x^{∗}(x_{i})|^{2})^{1/2} : ∥x^{∗}∥ ≤ 1} ,

so that π_{2}(T) ≤ B_{p}C_{2}(Y)π_{p}(T).

We prove next an important inequality, called Grothendieck’s inequality, which is an essential tool in studying p-summing operators as well as other topics. The inequality states: There is a universal constant K_{G} so that when-

ever A = (a_{ij})_{i,j=1}^{n} is a scalar matrix so that | ∑_{i,j=1}^{n} a_{ij}t_{i}s_{j}| ≤ 1 for all

{t_{i}}_{i=1}^{n} and {s_{j}}_{j=1}^{n} of absolute value ≤ 1 then | ∑_{i,j=1}^{n} a_{ij}⟨x_{i}, y_{j}⟩| ≤ K_{G}

for all choices of vectors {x_{i}}_{i=1}^{n} and {y_{j}}_{j=1}^{n} in the unit ball of a Hilbert space H. A more conceptual (but obviously equivalent) way of stating Grothendieck’s inequality is the following. Suppose that the matrix A = (a_{ij})_{i,j=1}^{n} has norm at most one when considered as an operator from ℓ_{∞}^{n} to ℓ_{1}^{n}. Let H be a Hilbert space. Then ∥A ⊗ I_{H}∥ ≤ K_{G} as an operator from ℓ_{∞}^{n}(H) to ℓ_{1}^{n}(H), where

(A ⊗ I_{H})(y_{1}, …, y_{n}) := (∑_{j=1}^{n} a_{1j}y_{j}, …, ∑_{j=1}^{n} a_{nj}y_{j})

for (y_{1}, …, y_{n}) in ℓ_{∞}^{n}(H).

To prove Grothendieck’s inequality fix n and a matrix A = (a_{ij})_{i,j=1}^{n} with ∥A∥ := ∥A : ℓ_{∞}^{n} → ℓ_{1}^{n}∥ = 1 as above, and put !A! := ∥A ⊗ I_{H} : ℓ_{∞}^{n}(H) → ℓ_{1}^{n}(H)∥. We have to find an estimate on !A! independent of n and A. Note first that for all u_{i} and v_{j} in a Hilbert space

| ∑_{i,j=1}^{n} a_{ij}⟨u_{i}, v_{j}⟩| ≤ !A! max_{i} ∥u_{i}∥ max_{j} ∥v_{j}∥.

Since all Hilbert spaces are created equal, we can use for H any (infinite dimensional or even 2n-dimensional) Hilbert space. For the purpose of proving Grothendieck’s inequality, some Hilbert spaces are more equal than others! We use for H the subspace of L_{2}(0, 1) mentioned in section 4 consisting just of functions having a Gaussian distribution with mean 0. To prove Grothendieck’s inequality it is enough to consider norm one vectors {x_{i}}_{i=1}^{n} and {y_{j}}_{j=1}^{n} in H. Given 0 < δ < 1/2 there is an M = M(δ) so that for any norm one function f in H, ∥f − f^{M}∥_{2} = δ where

f^{M}(t) = \{ \begin{matrix}f(t) & if |f(t)| ≤ M \\ M sign f(t) & if |f(t)| > M. \end{matrix}

Note that by our assumption on the matrix A, for any choice of functions f_{i} and g_{j} in L_{2}(0, 1) which are uniformly bounded by M,

|∑_{i,j} a_{ij}⟨f_{i}, g_{j}⟩| = ∫_{0}^{1} |∑_{i,j} a_{ij}f_{i}(t)\overline{g_{j}(t)}| dt ≤ M^{2}.

Hence

|∑_{i,j} a_{ij}⟨x_{i}, y_{j}⟩| ≤ |∑_{i,j} a_{i,j}⟨x_{i}^{M}, y_{j}^{M}⟩| + |∑_{i,j} a_{ij}⟨x_{i}, y_{j} − y_{j}^{M}⟩| +

+ |∑_{i,j} a_{ij}⟨x_{i} − x_{i}^{M}, y_{j}^{M}⟩|

≤ M^{2} + 2!A!δ.

Since the supremum of the left hand side (over all choices of x_{i} and y_{j}) is by definition !A!, we deduce that !A! ≤ \frac{M^{2}}{1−2δ}. This completes the shortest proof we know for Grothendieck’s inequality. With the optimal choice of δ this proof yields that K_{G} < 8.69 (or K_{G} < 8.55 if one exercises more care in the computation), but it is known that K_{G} < 1.79. Incidentally, although the best value for K_{G} is unknown, it is known that for complex scalars it is smaller than for real ones.

The first application of Grothendieck’s inequality is: Every bounded linear operator T from ℓ_{1} to ℓ_{2} is absolutely summing and π_{1}(T) ≤ K_{G}∥T ∥. Indeed, let {e_{j}}_{j=1}^{∞} be the unit vector basis for ℓ_{1} and let u_{i} = ∑_{j=1}^{m} a_{ij}e_{j} be n

vectors in ℓ_{1}^{m} for some m so that ∑_{i=1}^{n} |x^{∗}(u_{i})| ≤ 1 for every unit vector x^{∗} in ℓ_{1}^{∗}.

For any choice {s_{j}}_{j=1}^{m} of scalars of absolute value ≤ 1 let x^{∗} ∈ ℓ_{∞}^{m} be defined by x^{∗}(e_{j}) = s_{j}. Then for all {t_{i}}_{i=1}^{n} with |t_{i}| ≤ 1,

|∑_{j} ∑_{i} a_{ij}t_{i}s_{j}| ≤ ∑_{i} |t_{i}| |∑_{j} a_{ij}x^{∗}(e_{j})| ≤ ∑_{i} |x^{∗}(u_{i})| ≤ 1.

Hence, if y_{i} ∈ ℓ_{2} with ⟨T u_{i}, y_{i}⟩ = ∥T u_{i}∥ and ∥y_{i}∥ = 1 for all i, we get

∑_{i} ∥T u_{i}∥ = ∑_{i}⟨T u_{i}, y_{i}⟩ = ∑_{j} ∑_{i} a_{ij}⟨T e_{j}, y_{i}⟩ ≤ K_{G}∥T ∥.

By the local nature of the definition of the 1-summing norm it follows immediately that if X is an L_{1,λ} space and Y an L_{2,μ} space then for every T : X → Y, π_{1}(T) ≤ λμK_{G}∥T ∥.

As a second application of Grothendieck’s inequality, we prove a result alluded to in section 5: If T is an operator from a Banach lattice X into a Banach lattice Y, then for all {x_{i}}_{i=1}^{n} ⊂ X

∥(∑_{i=1}^{n} |T x_{i}|^{2})^{1/2}∥ ≤ K_{G}∥T ∥ ∥(∑_{i=1}^{n} |x_{i}|^{2})^{1/2}∥ . (33)

The first step in the proof is to use Grothendieck’s inequality to prove (33) when X = ℓ_{∞}^{m} and Y = ℓ_{1}^{m} for some m. Having done that, the case where X is a C(K) space and Y is an L_{1}(μ) spaces follows immediately by approximation. Finally, we use the lattice theory discussed in section 5 to reduce the general case to the case when X = C(K) and Y = L_{1}(μ).

Before proving (33), observe that since L_{1}(μ) spaces are 2-concave we can deduce the following from (33): If T is an operator from a C(K) space into an L_{1}(μ) space, then T is 2-summing and π_{2}(T) ≤ K_{G}∥T ∥. Since L_{p}(μ), 1 ≤ p ≤ 2, embeds isometrically into L_{1}(ν) for some ν, the local nature of the 2-summing norm gives: Every operator T from an L_{∞,λ} space into a subspace of a L_{p,τ} space, 1 ≤ p ≤ 2, is 2-summing with π_{2}(T) ≤ λτK_{G}∥T ∥.

We turn to the proof of (33) when T is an operator from ℓ_{∞}^{m} to ℓ_{1}^{m}. Let {e_{i}}_{i=1}^{m} be the unit vector basis for ℓ_{∞}^{m} and let x_{k} = ∑_{i=1}^{m} b_{ki}e_{i} be in ℓ_{∞}^{m} with

∥( ∑_{k=1}^{n} |x_{k}|^{2})^{1/2}∥_{∞} ≤ 1. Let {u_{i}}_{i=1}^{n} be orthonormal in some Hilbert space

H, set for 1 ≤ i ≤ m, y_{i} = ∑_{k=1}^{n} b_{ki}u_{k}, and define in ℓ_{∞}^{m}(H) the vector x̃ :=

(y_{1}, …, y_{m}). Then ∥x̃∥_{ℓ_{∞}^{m}(H)} = ∥( ∑_{k=1}^{n} |x_{k}|^{2})^{1/2}∥_{∞} ≤ 1. Similarly,

∥(∑_{k=1}^{n} |T x_{k}|^{2})^{1/2}∥_{1} = ∥(T ⊗ I_{H})x̃∥_{ℓ_{1}^{m}(H)}

and this last quantity is at most K_{G}∥T ∥ by (the conceptual form of) Grothendieck’s inequality. This gives (33) when X = ℓ_{∞}^{m}, Y = ℓ_{1}^{m} and hence also when X = C(K), Y = L_{1}(μ).

Suppose now that X and Y are general Banach lattices and T : X → Y

has norm one. Let x_{1}, …, x_{n} be in X with ∥( ∑_{i=1}^{n} |x_{i}|^{2})^{1/2}∥ = 1 and set

u = ( ∑_{i=1}^{n} |x_{i}|^{2})^{1/2}. Since we are interested only in estimating ∥( ∑_{i=1}^{∞} |T x_{i}|^{2})^{1/2}∥,

we can assume by replacing X by the (separable) sublattice generated by x_{1}, …, x_{n} that X is separable. The space T X is then separable, so we can similarly assume that Y is separable. Given any ϵ > 0, as seen in section 5 there is a strictly positive functional y^{∗} ∈ Y^{∗} with

y^{∗} (∑_{i=1}^{n} |T x_{i}|^{2})^{1/2} ≥ ∥(∑_{i=1}^{n} |T x_{i}|^{2})^{1/2}∥

and ∥y^{∗}∥ ≤ 1 + ϵ. As mentioned in section 5, the space X_{u} is lattice isometric to a C(K) space and Y_{y^{∗}} is lattice isometric to an L_{1}(μ) space. The natural lattice homomorphisms J_{1} : X_{u} → X, J_{2} : Y → Y_{y^{∗}} satisfy ∥J_{1}∥ = ∥u∥ = 1 and ∥J_{2}∥ = ∥y^{∗}∥ ≤ 1 + ϵ. We can then apply (33) to the operator J_{2}T J_{1} to obtain that

∥(∑_{i=1}^{n} |T x_{i}|^{2})^{1/2}∥_{Y_{y^{∗}}} ≤ K_{G}∥J_{2}T J_{1}∥ ∥(∑_{i=1}^{n} |x_{i}|^{2})^{1/2}∥_{X_{u}}
≤ (1 + ϵ)K_{G}∥T ∥.

Since

∥(∑_{i=1}^{n} |T x_{i}|^{2})^{1/2}∥_{Y_{y^{∗}}} = y^{∗} ((∑_{i=1}^{n} |T x_{i}|^{2})^{1/2}) ≥ ∥(∑_{i=1}^{n} |T x_{i}|^{2})^{1/2}∥ ,

this completes the proof.

We give now one of the many applications of the preceding results to the geometry of Banach spaces. Every space X which is isomorphic to a quotient space of an L_{∞} space and is as well a subspace of an L_{1} space must be isomorphic to a Hilbert space. Indeed, if we compose the map from an L_{∞} space into X with the embedding of X into an L_{1} space we get an operator from an L_{∞} space to an L_{1} space which is thus 2-summing. Like any 2-summing operator this operator factors through a Hilbert space. It follows that X is a quotient space of a Hilbert space and thus isomorphic to a Hilbert space.

A quantitative inspection of the argument above shows that if X is a quotient of a C(K) space and a subspace of an L_{1} space then d(X, ℓ_{2}) ≤ K_{G} (provided X is separable and infinite dimensional). Note that ℓ_{2} is isometric to a subspace of L_{1}(0, 1) (see section 4) and thus ℓ_{2}^{∗} = ℓ_{2} is also a quotient space of C(0, 1) because ℓ_{2} is reflexive and C(0, 1), considered as a subspace of L_{∞}(0, 1) = L_{1}(0, 1)^{∗}, determines the norm of L_{1}(0, 1). (If Y is a reflexive subspace of a Banach space X and Z is a norm closed subspace of X^{∗} which determines the norm of X, then the restriction operator z^{∗} ↦ z_{|Y}^{∗} is a quotient mapping from Z onto Y^{∗}.)

We will next give a glimpse into the connections of p-summing operators to the theory of finite dimensional spaces discussed in section 8. If X is the space ℓ_{2}^{n} then π_{2}(I_{X}) = \sqrt{n} since this is the Hilbert-Schmidt norm of the identity in ℓ_{2}^{n}. It is quite surprising that: For any X with dim X = n, π_{2}(I_{X}) = \sqrt{n}. To see this take any norm one operator V from ℓ_{2} into X. By projecting onto the orthogonal complement of the kernel of V, we can factor V as V_{1}P where V_{1} : ℓ_{2}^{m} → X, P : ℓ_{2} → ℓ_{2}^{m}, ∥P ∥ = 1, ∥V_{1}∥ = ∥V ∥, and m ≤ n. Then

π_{2}(I_{X}V) ≤ ∥I_{X}∥ ∥V_{1}∥π_{2}(I_{ℓ_{2}^{m}})∥P ∥ ≤ \sqrt{n}.

Taking the supremum over all such V gives π_{2}(I_{X}) ≤ \sqrt{n}.

To prove the other inequality, we get from the Pietsch factorization theorem a probability measure μ on B_{X^{∗}} and T : L_{2}(μ) → X so that T I_{2|_{X}} = I_{X} and ∥T ∥ = π_{2}(I_{X}), where as usual I_{2} : C(B_{X^{∗}}) → L_{2}(μ) is the formal identity and X is canonically embedded into C(B_{X^{∗}}). The space X_{2} := I_{2}X is an n-dimensional Hilbert space and I_{2}T is the identity on X_{2}. Hence

\sqrt{n} = π_{2}(I_{X_{2}}) ≤ π_{2}(I_{2})∥T ∥ = π_{2}(I_{X}).

As a consequence of the preceding we deduce: The projection constant of an n-dimensional space X is at most \sqrt{n}; that is, whenever Y contains X there is a projection P from Y onto X with ∥P ∥ ≤ \sqrt{n}. Indeed, since π_{2}(I_{X}) = \sqrt{n}, the Pietsch factorization theorem yields that the identity operator on X can be represented as X ⟶^{J} L_{∞}(μ) ⟶^{I_{2}} L_{2}(μ) ⟶^{V} X, with J an isometry, I_{2} the formal identity, and ∥V ∥ = \sqrt{n}. Since L_{∞}(μ) is 1-injective the operator J can be extended to a norm one operator T from Y into L_{∞}(μ). The operator P := V I_{2}T is a projection from Y onto X with norm at most \sqrt{n}. Notice also that the factorization used above gives another proof of the result proved in section 8 that d(X, ℓ_{2}^{n}) ≤ \sqrt{n}.

The estimate of \sqrt{n} for the projection constant of an n-dimensional space is essentially sharp. This is discussed in [33].

The p-integral operators form a class of operators which are closely related to the p-summing operators. An operator T : X → Y is said to be p-integral, 1 ≤ p ≤ ∞, (in symbols T ∈ I_{p}(X, Y)) provided that the composition J_{Y} T of T with the canonical embedding J_{Y} : Y → Y^{∗∗} factors through the formal identity I_{∞,p} : L_{∞}(μ) → L_{p}(μ) for some probability measure μ:

\begin{matrix}L_{∞}(μ) & ⟶^{I_{∞,p}} & L_{p}(μ) \\ _{A} ↑ & ↓ _{B} \\ X & ⟶^{T} & Y & ⟶^{J_{Y}} & Y^{∗∗} \end{matrix} (34)

The p-integral norm i_{p}(T) is then defined to be the infimum over all such factorizations of ∥A∥∥B∥. By taking ultraproducts one sees that this infimum is really a minimum. The space (I_{p}(X, Y), i_{p}) is easily seen to be a Banach space and i_{p} satisfies the ideal property i_{p}(ST U) ≤ ∥S∥i_{p}(T)∥U∥.

If T is in I_{p}(X, Y) and X is a subspace of C(K), with K a compact Hausdorff space, then there is a probability measure ν on K and an operator S : L_{p}(ν) → Y^{∗∗} with ∥S∥ = i_{p}(T) which makes the following diagram commute:

\begin{matrix}C(K) & ⟶^{I_{p}} & L_{p}(ν) \\ ∪ & ↓S \\ X & ⟶^{T} & Y & ⟶^{J_{Y}} & Y^{∗∗} \end{matrix} (35)

Indeed, if (34) holds, A can be extended to an operator Ã : C(K) → L_{∞}(μ) because L_{∞}(μ) is 1-injective. By the Pietsch factorization theorem there is a probability measure ν on K so that for each x ∈ C(K),

∥BI_{∞,p}Ãx∥ ≤ π_{p}(BI_{∞,p}Ã) (∫_{K} |x|^{p} dν)^{1/p}

and the desired conclusion follows from

π_{p}(BI_{∞,p}Ã) ≤ ∥B∥π_{p}(I_{∞,p})∥Ã∥ = ∥B∥∥A∥.

It is evident that π_{p}(T) ≤ i_{p}(T) and from the Pietsch factorization theorem it follows that π_{p}(T) = i_{p}(T) if the domain of T is a C(K) space. As was mentioned implicitly in the discussion of 2-summing operators, π_{2}(T) = i_{2}(T) for any operator. One reason for defining p-integral via a factorization of J_{Y} T rather than T is that this forces i_{p}(T) = i_{p}(T^{∗∗}) (use the fact that a dual space is norm one complemented in its bidual).

The 1-injectivity of C(K)^{∗∗} gives that a p-summing operator T into a C(K) space is p-integral with i_{p}(T) = π_{p}(T). The equality i_{1}(T^{∗}) = i_{1}(T) follows from the observation that the adjoint I_{1}^{∗} of I_{1} : C(K) → L_{1}(ν) is I_{∞,1} : L_{∞}(ν) → L_{1}(ν) followed by the identification of L_{1}(ν) with the norm one complemented subspace of C(K)^{∗} consisting of the finite signed measures which are absolutely continuous with respect to ν.

For other values of p, the adjoint of a p-integral operator need not be strictly singular (see [9, 5.12]) and hence need not be q-summing for any q < ∞.

For each 1 ≤ p ≤ ∞, p ≠ 2, there exist p-summing operators which are not p-integral (see [9, 5.13]). The case p = 1 is particularly easy to deduce from the theory we have presented. We saw that every operator T : ℓ_{1} → ℓ_{2} is 1-summing. If T : ℓ_{1} → ℓ_{2} is 1-integral, then T has a factorization ℓ_{1}⟶^{A}L_{∞}(μ)⟶^{I_{∞,1}}L_{1}(μ)⟶^{B}ℓ_{2}. But then B and also I_{∞,1} are 1-summing, hence BI_{∞,1}, whence also BI_{∞,1}A = T, are compact (use the fact that L_{1}(μ) has the DP property).

For p = ∞ and Y reflexive the infinity integral operators from X to Y are exactly those which factor through some L_{∞}(μ) space (with the integral norm equal to the best factorization). In particular for X reflexive the infinity integral norm of the identity of X is finite if and only if X is finite dimensional (and is equal to the projection constant of X in that case). Thus also for p = ∞ it is evident that summing and integral norms can be very different.

The main reason for introducing p-integral operators is that they are needed for the duality theory of Π_{p}(X, Y). For simplicity, we restrict to the case where X and Y are finite dimensional. Following the notation used in section 8, for finite dimensional X, Y and α a norm on B(X, Y) we represent the dual of (B(X, Y), α) as (B(Y, X), α^{∗}), where the pairing is given by ⟨S, T ⟩ = trace T S (= trace ST). Then for all 1 ≤ p ≤ ∞, Π_{p}(X, Y)^{∗} = I_{p^{∗}}(Y, X) when X and Y are finite dimensional. This just means that for each S ∈ B(Y, X), i_{p^{∗}}(S) = sup {trace T S : T ∈ B(X, Y), π_{p}(T) ≤ 1} .

In view of the discussion in section 8 that (B(X, Y), ∥ · ∥)^{∗} = N (Y, X) for finite dimensional spaces, we have (since π_{∞} = ∥ · ∥) that the identity i_{1}(S) = N (S) for operators between finite dimensional spaces is equivalent to the assertion that π_{∞}^{∗} = i_{1}. To prove that i_{1}(S) = N (S), first note (it is more or less immediate from the definition of N (S)) that N (S) is the infimum of ∥A∥∥Δ∥∥B∥ over all factorizations of S of the form

\begin{matrix}ℓ_{∞}^{n} & ⟶^{Δ} & ℓ_{1}^{n} \\ _{A} ↑ & & ↓ _{B} \\ Y & ⟶^{S} & X \end{matrix} (36)

with Δ a diagonal operator. From this it follows easily that i_{1}(S) ≤ N (S). For the reverse inequality, given a factorization

\begin{matrix}L_{∞}(μ) & ⟶^{I_{∞,1}} & L_{1}(μ) \\ _{A_{1}} ↑ & & ↓ _{B_{1}} \\ Y & ⟶^{S} & X \end{matrix} (37)

one has for ϵ > 0 an operator A_{ϵ} : Y → L_{∞}(μ) with ∥A_{1}−A_{ϵ}∥ < ϵ so that A_{ϵ}Y is contained in the simple functions. This gives a factorization of B_{1}I_{∞,1}A_{ϵ} of the form (36) with ∥B∥∥Δ∥∥A∥ ≤ ∥B_{1}∥∥I_{∞,1}∥∥A_{ϵ}∥. Setting N := dim Y, we get N (S − B_{1}I_{∞,1}A_{ϵ}) ≤ N∥S − B_{1}I_{∞,1}A_{ϵ}∥ ≤ N∥A_{1} − A_{ϵ}∥ < Nϵ. Letting ϵ → 0, we conclude that N (S) ≤ i_{1}(S).

Having seen that i_{1}(S) = N (S) for operators between finite dimensional spaces, one can easily deduce the identity π_{p}^{∗} = i_{p^{∗}} for 1 < p < ∞ from the composition inequality i_{1}(ST) ≤ π_{p}(T)i_{p^{∗}}(S). In fact, a more general inequality is true for operators between general spaces: If T ∈ I_{p}(X, Y) and S ∈ Π_{q}(Y, Z) then ST ∈ I_{r}(X, Z) and i_{r}(ST) ≤ i_{p}(T)π_{q}(S), where \frac{1}{r} = 1 ∧ (\frac{1}{p} + \frac{1}{q}). This inequality follows from the composition inequality for p-summing operators mentioned in section 10. Indeed, consider the following commutative diagram:

C(K) ⟶^{I_{p}} L_{p}(μ)
A ↑  ↓ _{B}
X ⟶^{T} Y ⟶^{J_{Y}} Y^{∗∗}  ⟶^{S^{∗∗}} Z^{∗∗}

The operator S^{∗∗}BI_{p} satisfies

i_{r}(S^{∗∗}BI_{p}) = π_{r}(S^{∗∗}BI_{p}) ≤ π_{p}(BI_{p})π_{q}(S^{∗∗}) ≤ ∥B∥π_{p}(I_{p})π_{q}(S) = ∥B∥π_{q}(S)

and i_{r}(J_{Z}ST) = i_{r}(S^{∗∗}J_{Y} T) = i_{r}(S^{∗∗}BI_{p}A) ≤ ∥B∥π_{q}(S)∥A∥.

The desired composition inequality follows by taking the infimum over pintegral factorizations of T.

A finite dimensional Banach space X is said to have enough symmetries provided that the only operators on X which commute with each isometry on X are the scalar multiples of the identity operator. Finite dimensional spaces with a 1-symmetric basis are examples of spaces with enough symmetries. On the other hand, B(ℓ_{2}^{n}, ℓ_{2}^{n}) has enough symmetries but, as noted in section 9, does not even have a “good” unconditional basis.

In general, ideal norms for the operators on a space with enough symmetries are better behaved than on general finite dimensional spaces. For example, if X has enough symmetries, the operator one obtains from applying Lewis’ lemma to an ideal norm on B(X, X) is a scalar multiple of I_{X}. This is a consequence of the following: If dim X = n and X has enough symmetries and α is an ideal norm, then α(I_{X})α^{∗}(I_{X}) = n. The inequality α(I_{X})α^{∗}(I_{X}) ≥ trace I_{X} = n is clear. To see the reverse inequality, take T ∈ B(X, X) so that trace T = n and α(I_{X})α^{∗}(T) = n. Let G be the (compact) group of isometries of X and μ normalized Haar measure on G. For each S in G, trace S^{−1}T S = trace T = n and, by the ideal property of α^{∗}, α^{∗}(S^{−1}T S) = α^{∗}(T). The operator

T_{0} := ∫_{G} S^{−1}T Sdμ(S)

satisfies trace T_{0} = trace T, α^{∗}(T_{0}) ≤ α^{∗}(T), and T commutes with all elements of G. Hence T_{0} = λI_{X} and since trace T_{0} = n we conclude that λ = 1. It follows that α(I_{X})α^{∗}(I_{X}) ≤ n and thus we have equality.

A particular case of this last result is worth singling out: If dim X = n and X has enough symmetries then the projection constant of X is equal to n/π_{1}(I_{X}).

11 Interpolation

In this section we give a very sketchy introduction to modern interpolation theory and mention a few applications to Banach space theory. The basic theory is exposed in the books [4] as well as [13] and [5]. The article [31] gives an overview of the subject and further connections to Banach space theory.

We begin by describing the abstract framework of interpolation theory. A Banach couple is a pair \overline{X} = (X_{0}, X_{1}) of Banach spaces which are contained in a Hausdorff topological vector space Z such that both injections J_{i}: X_{i} → Z, i = 0, 1, are continuous. The ambient space Z can be replaced with the algebraic sum Σ(\overline{X}) := X_{0} + X_{1} ⊂ Z topologized with the (complete) norm

∥x∥ = inf{∥x_{0}∥_{[0]} + ∥x_{1}∥_{[1]}: x = x_{0} + x_{1}, x_{0} ∈ X_{0}, x_{1} ∈ X_{1}}

where ∥ · ∥_{[i]} is the norm of X_{i}. We always take for Z the space Σ(\overline{X}). The space Δ(\overline{X}) := X_{0} ∩ X_{1} also plays a role in the theory; it is naturally normed by the (complete) norm

∥x∥ = ∥x∥_{[0]} ∨ ∥x∥_{[1]}.

Thus we have for i = 0, 1 inclusions

Δ(\overline{X}) ↪^{H_{i}} X_{i} ↪^{J_{i}} Σ(\overline{X})

with ∥H_{i}∥ ≤ 1, ∥J_{i}∥ ≤ 1. Any Banach space X satisfying Δ(\overline{X}) ⊂ X ⊂ Σ(\overline{X}) with both inclusions continuous is called an intermediate space between X_{0} and X_{1} (or an intermediate space with respect to \overline{X}). The spaces X_{0}, X_{1}, Δ(\overline{X}), and Σ(\overline{X}) are all intermediate spaces with respect to \overline{X}.

Given Banach couples \overline{X} = (X_{0}, X_{1}), \overline{Y} = (Y_{0}, Y_{1}), and a linear mapping T: Σ(\overline{X}) → ∑(\overline{Y}), we write T ∈ B(\overline{X}, \overline{Y}) provided T_{|X_{i}} ∈ B(X_{i}, Y_{i}) for i = 0, 1.

If X and Y are intermediate spaces with respect to \overline{X} and \overline{Y}, respectively, we say that X and Y are an interpolation pair for \overline{X} and \overline{Y} provided that if T ∈ B(\overline{X}, \overline{Y}), then T_{|X} ∈ B(X, Y). If always

∥T ∥_{X,Y} ≤ ∥T ∥_{X_{0},Y_{0}} ∨ ∥T ∥_{X_{1},Y_{1}}

X and Y are said to be an exact interpolation pair for \overline{X} and \overline{Y}, where ∥S∥_{X,Y} := ∥S_{|X}∥_{B(X,Y)} (if SX ⊄ Y then ∥S∥_{X,Y} := ∞). Finally, if 0 ≤ θ ≤ 1 is such that the inequality

∥T ∥_{X,Y} ≤ C∥T ∥_{X_{0},Y_{0}}^{1−θ} · ∥T ∥_{X_{1},Y_{1}}^{θ}

always holds, then the interpolation pair X and Y are said to be of exponent θ (and exact of exponent θ if C = 1). When \overline{X} = \overline{Y} and X = Y, we abbreviate by saying e.g. that X is an interpolation space with respect to \overline{X} provided X and X are an interpolation pair with respect to \overline{X} and \overline{X}.

The most classical realization of this abstract set-up occurs in the scale of L_{p}(μ) spaces. For 1 ≤ p_{0}, p_{1} ≤ ∞, the pair (L_{p_{0}}(μ), L_{p_{1}}(μ)) is a Banach couple, and for p_{0} ∧ p_{1} ≤ p ≤ p_{0} ∨ p_{1} the space L_{p}(μ) is an intermediate space between L_{p_{0}}(μ) and L_{p_{1}}(μ). In the language of interpolation theory the Riesz– Thorin interpolation theorem can be stated as follows. Let 1 ≤ p_{0}, p_{1} ≤ ∞, 1 ≤ q_{0}, q_{1} ≤ ∞, and 0 < θ < 1. Define p and q by

\frac{1}{p} = \frac{1 − θ}{p_{0}} + \frac{θ}{p_{1}} , \frac{1}{q} = \frac{1 − θ}{q_{0}} + \frac{θ}{q_{1}} .

Then for all measures μ and ν, (complex) L_{p}(μ) and L_{q}(ν) are an exact interpolation pair of exponent θ with respect to (L_{p_{0}}(μ), L_{p_{1}}(μ)) and (L_{q_{0}}(ν), L_{q_{1}}(ν)). It is formal to derive from the Riesz–Thorin theorem that its statement remains true in the setting of real scalars as long as p_{0} ≤ p_{1} and q_{0} ≤ q_{1} (or, what is the same, if p_{1} ≤ p_{0} and q_{1} ≤ q_{0}). In other cases the word “exact” must be deleted from the statement of the Riesz–Thorin theorem when the scalars are real.

The modern approach to the Riesz–Thorin theorem proceeds via a construction called the complex method. Let \overline{X} = (X_{0}, X_{1}) be a Banach couple of complex Banach spaces and let H(\overline{X}) denote the space of functions f from the strip S := \{z ∈ ℂ: 0 ≤ Re z ≤ 1\} into Σ(\overline{X}) which are bounded, continuous, and analytic in the interior S_{0} of S, and so that for j = 0, 1 the restriction of f to the line j + iℝ is a bounded, continuous function into X_{[j]}. The space H(\overline{X}) is normed by

∥f∥_{H(X)} = sup_{t∈ℝ} ∥f(it)∥_{[0]} ∨ ∥f(1 + it)∥_{[1]}.

The norm ∥ · ∥_{H(\overline{X})} is a complete norm on H(\overline{X}) ([4, 4.1.1]). Indeed, by the maximum modulus principle, for z ∈ S

∥f(z)∥_{Σ(\overline{X})} ≤ sup_{t∈ℝ} ∥f(it)∥_{Σ(\overline{X})} ∨ ∥f(1 + it)∥_{Σ(\overline{X})} ≤ ∥f∥_{H(\overline{X})}.

Using the completeness of Σ(\overline{X}), it is easy to check that if ∑_{n=1}^{∞} ∥f_{n}∥_{H(\overline{X})} < ∞ then ∑_{n=1}^{∞} f_{n} converges in H(\overline{X}).

For 0 < θ < 1 define \overline{X}_{[θ]} (or simply X_{[θ]}) to be all x in Σ(\overline{X}) for which x = f(θ) for some f ∈ H(\overline{X}) with the norm

∥x∥_{[θ]} := inf\{∥f∥_{H(\overline{X})}: f(θ) = x\} < ∞.

The space (X_{[θ]}, ∥·∥_{[θ]}) is in a natural way isometric to the quotient H(\overline{X})/N_{θ}(\overline{X}), where N_{θ}(X) := \{f ∈ H(\overline{X}): f(θ) = 0\} is the kernel of the operator H(\overline{X}) → ∑(\overline{X}) defined by f ↦ f(θ), which as noted above has norm at most one. Consequently X_{[θ]} ⊂ Σ(\overline{X}) with the inclusion of norm ≤ 1. To see that Δ(\overline{X}) ⊂ X_{[θ]} with inclusion of norm ≤ 1, take any x in Δ(\overline{X}) and define f ∈ H(\overline{X}) by f(z) ≡ x. Then f(θ) = x and for t ∈ ℝ we have

∥f(it)∥_{[0]} = ∥x∥_{[0]}, ∥f(1 + it)∥_{[1]} = ∥x∥_{[1]}

so that

∥x∥_{[0]} ≤ ∥f∥_{H(\overline{X})} ≤ ∥x∥_{[0]} ∨ ∥x∥_{[1]} = ∥x∥_{Δ(\overline{X})}.

Thus X_{[θ]} is an intermediate space with respect to \overline{X}. Moreover, if \overline{X} and \overline{Y} are Banach couples and 0 < θ < 1, then X_{[θ]} and Y_{[θ]} are an exact interpolation pair of exponent θ with respect to \overline{X} and \overline{Y}. To see this, let T ∈ B(\overline{X}, \overline{Y}), x ∈ X_{[θ]}, ε > 0, and choose f ∈ H(\overline{X}) so that f(θ) = x with ∥f∥_{H(\overline{X})} ≤ ∥x∥_{[θ]} + ε. Define g: S → ∑(\overline{Y}) by

g(z) = ∥T ∥_{X_{0},Y_{0}}^{z−1}∥T ∥_{X_{1},Y_{1}}^{−z}T f(z).

Then g is in H(\overline{Y}), ∥g∥_{H(\overline{Y})} ≤ ∥f∥_{H(\overline{X})}, and g(θ) = ∥T ∥_{X_{0},Y_{0}}^{θ−1}∥T ∥_{X_{1},Y_{1}}^{−θ}T x so that

\begin{aligned}∥T x∥_{[θ]} & ≤ & ∥T ∥_{X_{0},Y_{0}}^{1−θ}∥T ∥_{X_{1},Y_{1}}^{θ}∥g∥_{H(\overline{Y})} \\ & ≤ & ∥T ∥_{X_{0},Y_{0}}^{1−θ}∥T ∥_{X_{1},Y_{1}}^{θ}(∥x∥_{[θ]} + ε). \end{aligned}

In order to derive the Riesz–Thorin theorem from this last result, it suffices to check that (L_{p_{0}}(μ), L_{p_{1}}(μ))_{[θ]} can be identified with L_{p}(μ), where \frac{1}{p} = \frac{1−θ}{p_{0}} + \frac{θ}{p_{1}} . First suppose that x ∈ L_{p}(μ) with ∥x∥_{p} = 1. Define f ∈ H := H(L_{p_{0}}(μ), L_{p_{1}}(μ)) by

f(z) = |x|^{p/p(z)} \frac{x}{|x|}

where \frac{1}{p(z)} := \frac{(1−z)}{p_{0}} + \frac{z}{p_{1}} and \frac{0}{0} := 0. Evidently ∥f∥_{H} ≤ 1 and f(θ) = x, so x ∈ (L_{p_{0}}(μ), L_{p_{1}}(μ))_{[θ]} and ∥x∥_{[θ]} ≤ 1 = ∥x∥_{p}.

Next suppose that x ∈ (L_{p_{0}}(μ), L_{p_{1}}(μ))_{[θ]}, ∥x∥_{[θ]} < 1, and f ∈ H satisfies ∥f∥_{H} < 1, f(θ) = x. To see that ∥x∥_{p} ≤ 1 it is sufficient to check that if y is a μ-measurable function with μ[y ≠ 0] < ∞ and ∥y∥_{p^{∗}} ≤ 1, then |⟨x, y⟩| := ∫ |xy|dμ ≤ 1. For z ∈ S set

g(z) = |y|^{p^{∗}/p^{∗}(z)} \frac{y}{|y|},

where \frac{1}{p^{∗}(z)} := \frac{1−z}{p_{0}^{∗}} + \frac{z}{p_{1}^{∗}}, and define h(z) = ⟨f(z), g(z)⟩. Then since ∥f∥_{H} < 1, for t ∈ ℝ we have |h(it)| ∨ |h(1 + it)| ≤ 1. By the maximum modulus principle we conclude that |⟨x, y⟩| = |h(θ)| ≤ 1 and hence |⟨x, y⟩| ≤ 1.

Of the many classical applications of the Riesz–Thorin theorem we mention only one, the Hausdorff–Young theorem: Let F be the Fourier transform for functions on ℝ^{n}, defined by Ff(s) = ∫_{ℝ^{n}} f(t)e^{−i⟨t,s⟩}dt. Then F maps L_{p}(ℝ^{n}) into L_{p^{∗}}(ℝ^{n}) for 1 ≤ p ≤ 2. To prove the Hausdorff–Young theorem, just observe that ∥F∥_{L_{1},L_{∞}} = 1 and ∥F∥_{L_{2},L_{2}} = 2π^{\frac{n}{2}} , so that for 1 ≤ p ≤ 2, ∥F∥_{L_{p},L_{p^{∗}}} ≤ 2^{\frac{2}{p^{∗}}} π^{\frac{n}{p^{∗}}} by the Riesz–Thorin theorem.

There are also useful vector-valued versions of the Riesz–Thorin theorem. These can be deduced from the following. Let (X_{0}, X_{1}) be a Banach couple, 1 ≤ p_{0}, p_{1} < ∞, 0 < θ < 1, and set \frac{1}{p} := \frac{1−θ}{p_{0}} + \frac{θ}{p_{1}} . Then

(L_{p_{0}}(μ, X_{0}), L_{p_{1}}(μ, X_{1}))_{[θ]} = L_{p}(μ, (X_{0}, X_{1})_{[θ]}).

If p_{0} < ∞, then also

(L_{p_{0}}(μ, X_{0}), L_{∞}^{0}(μ, X_{1}))_{[θ]} = L_{p}(μ, X_{0}, X_{1})_{[θ]},

where L_{∞}^{0}(μ, X) is the closure in L_{∞}(μ, X) of the simple functions. The proof of this theorem (see [4, 5.1.2]) is only a bit more complicated than the argument given above that (L_{p_{0}}(μ), L_{p_{1}}(μ))_{[θ]} = L_{p}(μ). The argument does use the following general fact about the complex method. For any Banach couple \overline{X} and 0 < θ < 1, Δ(\overline{X}) is dense in \overline{X}_{[θ]} (see [4, 4.2.2]).

The identification L_{p} = (L_{p_{0}}, L_{p_{1}})_{[θ]}, \frac{1}{p} = \frac{1−θ}{p_{0}} + \frac{θ}{p_{1}} , has a generalization to Banach lattices. Assume that X_{0} and X_{1} are complex Banach lattices of μmeasurable functions which are ideals in the space of all μ-measurable functions, 0 < θ < 1, and set

X_{0}^{1−θ}X_{1}^{θ} := {(sign x_{0}x_{1})|x_{0}|^{1−θ}|x_{1}|^{θ} : x_{0} ∈ X_{0}, x_{1} ∈ X_{1}}.

If X_{0} is order continuous then (X_{0}, X_{1})_{[θ]} = X_{0}^{1−θ}X_{1}^{θ} and

∥x∥_{[θ]} = inf{∥x_{0}∥_{[0]}^{1−θ}∥x_{1}∥_{[1]}^{θ}: |x| = |x_{0}|^{1−θ}|x_{1}|^{θ}}. (38)

For a proof see section iv.1.11 in [13]. The inequality ≤ in (38); namely,

∥|x_{0}|^{1−θ}|x_{1}|^{θ}∥_{[θ]} ≤ ∥x_{0}∥_{[0]}^{1−θ}∥x_{1}∥_{[1]}^{θ},

does not require order continuity of X_{0}. It reduces to the lattice Hölder inequality (4) when X_{0} = X_{1}.

Since X_{0}^{1−θ}L_{∞}(μ)^{θ} = X_{0}^{1−θ} = X_{0}^{( \frac{1}{1−θ})} (the \frac{1}{1−θ}-convexification of X_{0}), it follows that the p-convexification, 1 < p < ∞, of an order continuous Banach lattice X_{0} can be obtained by using the complex method to interpolate X_{0} with an appropriate abstract M-space (here there is no problem in passing to the complexification of a real Banach lattice and returning at the end to the real setting).

We turn now to a discussion of another interpolation method, which is a special case of what is usually called the K-method. Given a Banach couple \overline{X} = (X_{0}, X_{1}) and 0 < a, b < ∞, let K(·, a, b) be the norm of the algebraic sum of (X_{0}, a∥ · ∥_{[0]}) and (X_{1}, b∥ · ∥_{[1]}); that is,

K(x, a, b) = inf{a∥x_{0}∥_{[0]} + b∥x_{1}∥_{[1]}: x = x_{0} + x_{1}, x_{0} ∈ X_{0}; x_{1} ∈ X_{1}}. (39)

Here we follow the notation of [15, 2.g] because there applications of the Kmethod to Banach space theory are described. Usually in interpolation theory one uses only K(·, t) := K(·, 1, t); in this notation K(·, a, b) is just aK(·, \frac{b}{a}).

Let \overline{X} = (X_{0}, X_{1}) be a Banach couple. We mention a method for using the K-functional to build interpolation spaces with respect to \overline{X}. Let E be a space with a normalized 1-unconditional basis {e_{n}}_{n=1}^{∞} and let a = {a_{n}}_{n=1}^{∞}, b = {b_{n}}_{n=1}^{∞}, be sequences of positive scalars such that ∑_{n=1}^{∞} a_{n} ∧ b_{n} < ∞. The

space K(\overline{X}, E, a, b) is defined to be all x ∈ Σ(\overline{X}) for which ∑_{n=1}^{∞} K(x, a_{n}, b_{n})e_{n}

converges in E, normed by

∥x∥_{K(\overline{X},a,b)} = ∥∑_{n=1}^{∞} K(x, a_{n}, b_{n})e_{n}∥_{E} . (40)

It is simple to check that if \overline{Y} is another Banach couple, then K(\overline{X}, E, a, b) and K(\overline{Y} , E, a, b) are an exact interpolation pair with respect to \overline{X} and \overline{Y}.

Earlier we saw how the complex method can be used to construct the pconvexification of a Banach lattice. The K-method can also be used to construct Banach spaces with interesting properties. Suppose, for example, that both X_{0} and X_{1} have a normalized 1-unconditional basis, each of which we identify with the unit vectors {e_{n}}_{n=1}^{∞} so that X_{0} and X_{1} are both contained in c_{0} and thus \overline{X} = (X_{0}, X_{1}) is a Banach couple. Let E be another space with a normalized 1-unconditional basis, also denoted by \{e_{n}\}_{n=1}^{∞}. Since K(\overline{X}, E, a, b) is an exact interpolation space with respect to \overline{X}, it follows that \{e_{n}\}_{n=1}^{∞} is a 1-unconditional basis for K(\overline{X}, E, a, b) which moreover is even 1-symmetric if {e_{n}}_{n=1}^{∞} is 1-symmetric in both X_{0} and in X_{1} (one need only check that the span of \{e_{n}\}_{n=1}^{∞} is dense in K(\overline{X}, E, a, b)). Assume now that \{e_{n}\}_{n=1}^{∞} is 1-symmetric in both X_{0} and X_{1}, X_{0} ⊂ X_{1}, and ∥∑_{k=1}^{n} e_{k}∥_{[0]}^{−1} ∥∑_{k=1}^{n} e_{k}∥_{[1]} → 0 as n → ∞. It turns out (see [14, 3.b.4]) that if a_{n} = b_{n}^{−1} and the weight sequence b satisfies b_{n} ↑ ∞ sufficiently quickly, then there are disjoint finite subsets C_{n} of ℕ so that the mapping e_{n} ↦ ∥1_{C_{n}}∥_{K(\overline{X},a,b)}1_{C_{n}} extends to an isomorphism from E onto a complemented subspace of K(\overline{X}, E, a, b). This is how one proves the result mentioned in section 3 that a space with an unconditional basis is isomorphic to a complemented subspace of a space which has a symmetric basis.

“Good” properties (such as reflexivity and superreflexivity) possessed by the Banach spaces X_{0} and X_{1} (often possessed by just one of the spaces) generally pass to interpolation spaces between X_{0} and X_{1} which are obtained by the complex method or by the K-method (at least when the weight sequences a, b satisfy some growth conditions and the space E is “nice”). Sometimes interpolation spaces even have a good property which neither X_{0} nor X_{1} possesses. For example, suppose that X_{0} ⊂ X_{1} (this can be relaxed but is good enough for applications). If E is reflexive, the inclusion J: X_{0} → X_{1} is weakly compact, ∑_{n=1}^{∞} a_{n} < ∞, and b_{n} ↑ ∞, then K(\overline{X}, E, a, b) is reflexive. For

a proof when E = ℓ_{2}, which is easily modified to cover the general case, see [15, 2.g.11]. A consequence of this result is the following factorization theorem for weakly compact operators. If T : X → X_{1}, is weakly compact then T factors through a reflexive space; that is, there exists a reflexive Banach space Y and operators A ∈ B(X, Y), B ∈ B(Y, X_{1}) so that T = BA. To derive this factorization theorem from the interpolation result mentioned above, it suffices to take Y = K(X_{0}, X_{1}, ℓ_{2}, {2^{−n}}_{n=1}^{∞}, {2^{n}}_{n=1}^{∞}) where X_{0} is the span in X of \overline{T B_{X}} with \overline{T B_{X}} as the unit ball of X_{0}. Let A be T, considered as an operator into Y, and let B be the formal inclusion from Y into X_{1}. Then A and B are operators and T = BA.

We turn now to the realization of the K-method which is used most often in analysis and is discussed extensively in the books on interpolation theory we have mentioned. Given a Banach couple \overline{X} = (X_{0}, X_{1}), 0 < θ < 1, and 1 ≤ p < ∞, \overline{X}_{θ,p} denotes the space K(\overline{X}, ℓ_{p}, a, b) where a_{2n} := e^{θn}, b_{2n} := e^{−(1−θ)n}, a_{2n+1} := e^{−θn}, b_{2n+1} := e^{(1−θ)n}. Instead of using ∥ · ∥_{K(\overline{X},a,b)} on X_{θ,p}, it is customary to use

∥x∥_{θ,p} = inf ( _{ℝ}^{∫} ∥e^{θt}x_{0}(t)∥_{[0]}^{p}dt)^{1/p} ∨ ( _{ℝ}^{∫} ∥e^{−(1−θ)t}x_{1}(t)∥_{[1]}^{p}dt)^{1/p} (41)

where the infimum is over all x_{0}(t), x_{1}(t) for which e^{θt}x_{0}(t) ∈ L_{p}(ℝ, X_{0}), e^{−(1−θ)t}x_{1}(t) ∈ L_{p}(ℝ, X_{1}), and x = x_{0}(t)+x_{1}(t) for every t in ℝ. The expression ∥ · ∥_{θ,p} is a norm on \overline{X}_{θ,p} which is equivalent to ∥ · ∥_{K(\overline{X},a,b)} but is better behaved. For example, (\overline{X}_{θ,p}, ∥ · ∥_{θ,p}) is uniformly convex (rather than just superreflexive) if either X_{0} or X_{1} is uniformly convex and 1 < p < ∞ ([15, 2.g.21]). Moreover, ∥ · ∥_{θ,p} is very good for interpolation purposes, for if \overline{Y} is another Banach couple then (\overline{X}_{θ,p}, ∥ · ∥_{θ,p}) and (\overline{Y}_{θ,p}, ∥ · ∥_{θ,p}) are an exact interpolation pair of exponent θ with respect to \overline{X} and \overline{Y}.

It is of course important to identify \overline{X}_{θ,p} when \overline{X} is a concrete Banach couple. The spaces that arise in this connection when \overline{X} is a couple of L_{p}(μ) spaces are the L_{p,q} spaces. Given 0 < p < ∞, 0 < q < ∞, a measure μ, and a Banach space X, L_{p,q}(μ, X) is the space of X valued strongly measurable functions x for which

∥x∥_{p,q} := (q/p ∫_{0}^{∞}|t^{1/p}x^{∗}(t)|^{q} \frac{dt}{t})^{1/q} < ∞, (42)

where x^{∗}(t) is the decreasing rearrangement of ∥x(t)∥_{X}. For 0 < p ≤ ∞, L_{p,∞}(μ, X) is the space of X valued strongly measurable functions for which

∥x∥_{p,∞} := sup_{t>0} t^{1/p}x^{∗}(t) < ∞. (43)

When X is the scalar field we write L_{p,q}(μ). Evidently L_{p,p}(μ, X) = L_{p}(μ, X). Note that for p > q ≥ 1, the space L_{p,q}(μ) is the Lorentz function space L_{W,q}(μ) defined in section 5 with W (t) = qt^{q/p−1} and hence is a Banach space. When q > p ≥ 1 the expression ∥ · ∥_{p,q} does not satisfy the triangle inequality, although ∥ · ∥_{p,q} is equivalent to a norm when p > 1. In any case L_{p,q}(μ, X) is a metrizable topological vector space.

The main result about spaces obtained from L_{p} and L_{p,q} spaces via the Kmethod is the following (see [4, 5.3.1]): Let 0 < p_{0}, p_{1}, q_{0}, q_{1} ≤ ∞, 0 < θ < 1, and define p by \frac{1}{p} = \frac{1−θ}{p_{0}} + \frac{θ}{p_{1}} . If p_{0} ≠ p_{1}, or if p_{0} = p_{1} and \frac{1}{q} = \frac{1−θ}{q_{0}} + \frac{θ}{q_{1}} , then

(L_{p_{0},q_{0}}(μ, X), L_{p_{1},q_{1}}(μ, X))_{θ,q} = L_{p,q}(μ, X) (44)

up to equivalence of the “norms”; i.e., for some constant C > 0, C^{−1}∥ · ∥_{p,q} ≤ ∥·∥_{θ,q} ≤ C∥·∥_{p,q}. When the expressions ∥·∥_{p_{i},q_{i}}, i = 0, 1, are equivalent to norms one can deduce from (44) and earlier comments an interpolation theorem for operators. In fact, there are cases where interpolation is valid even when the spaces are not all Banach spaces. In particular, assume that T: L_{p_{i},r_{i}}(μ, X) → L_{q_{i},s_{i}}(ν, X) is continuous for i = 0, 1, with p_{0} ≠ p_{1}, q_{0} ≠ q_{1}, 0 < θ < 1, and define p, q by \frac{1}{p} = \frac{1−θ}{p_{0}} + \frac{θ}{p_{1}} , \frac{1}{q} = \frac{1−θ}{q_{0}} = \frac{θ}{q_{1}} . If p ≤ q, then T: L_{p}(μ, X) → L_{q}(ν, Y) is continuous and for 0 ≤ r ≤ s ≤ ∞, T: L_{p,r}(μ, X) → L_{q,s}(ν, Y) is continuous (see [4, 5.3.1]).

In some cases it is possible to characterize the interpolation spaces with respect to a concrete Banach couple. The Banach couple \overline{X} = (X_{0}, X_{1}) is called a Calderón couple if whenever x, y in Σ(\overline{X}) satisfy K(y, 1, t) ≤ K(x, 1, t) for all t, then there is T in B(\overline{X}, \overline{X}) so that T x = y. If always T can be chosen so that ∥T ∥_{X_{0},X_{0}} ∨ ∥T ∥_{X_{1},X_{1}} ≤ 1, then \overline{X} is called an exact Calderón couple. It is easy to see that if X is an intermediate space with respect to a Calderón couple \overline{X}, then X is an interpolation space with respect to \overline{X} if and only if it is K-monotone for \overline{X}, which means that if x ∈ X, y ∈ Δ(\overline{X}), and K(y, 1, t) ≤ K(x, 1, t) for all t, then y ∈ X. It then follows from the K-divisibility theorem that X is obtainable from \overline{X} via the K-method. This theorem says the following. If X is K-monotone for \overline{X}, then there are E, a, and b so that X = K(\overline{X}, E, a, b), up to an equivalent renorming.

The pair (L_{p}(0, 1), L_{q}(0, 1)), 1 ≤ p < q ≤ ∞, is a Calderón couple, and many other examples are known (see [31]). One interesting problem which is not completely solved is to determine a necessary and sufficient condition for a Banach couple of symmetric lattice ideals on [0, 1] (see section 5) to be a Calderón couple. Much is known about the interpolation spaces for such a couple. For example, an exact interpolation space with respect to such a couple must itself be a symmetric lattice ideal (the first step is to observe that if τ : [0, 1] → [0, 1] is a measure preserving automorphism, then x ↦ x(τ) defines an isometric automorphism of any symmetric lattice on [0, 1]). When the couple is (L_{1}(0, 1), L_{∞}(0, 1)), it is reasonable to guess that the converse is true. It is not, but: If L_{∞}(0, 1) is dense in the symmetric lattice ideal X, then X is an exact interpolation space with respect to (L_{1}(0, 1), L_{∞}(0, 1)) ([13, Th. 4.10]). Most natural symmetric lattice ideals on [0, 1], including all the separable ones, satisfy the hypothesis of this theorem.

12 List of symbols

Here is a list of symbols used in this introductory article and, where appropriate, a reference to where they are defined.

ℕ The natural numbers. ℝ The real numbers. ℂ The complex numbers. S̃ The complement of the set S. 𝕋 The unit circle in the complex plane. ℙ A probability measure (section 2). 𝔼 ∫ · dℙ (section 2). A_{P}, B_{p} The constants in Khintchine’s inequality (1); also the constants in

the Kahane-Khintchine inequality (24). K_{G} The constant in Grothendieck’s inequality (section 10). C(K; X) Continuous functions f on the (usually) compact Hausdorff space

K taking values in the (usually) normed space X, normed by ∥f∥ = sup_{t∈K} ∥f(t)∥.

C(K) C(K; X) when X is the scalar field. L_{p}(μ, X) The μ-measurable X-valued functions f for which

∥f∥_{p} := (∫ ∥f∥^{p} dμ)^{1/p} < ∞ (section 7). Here 0 < p < ∞. L_{p}(μ) L_{p}(μ, X) when X = ℝ. L_{∞}(μ, X) The μ-measurable essentially bounded X-valued functions, with

norm ∥f∥_{∞} := inf_{μA=0} sup |f_{|Ã}|. L_{p}(0, 1) L_{p}(μ) when μ is Lebesgue measure on the unit interval. L_{p}(𝕋) L_{p}(μ) when μ is normalized Lebesgue measure on the unit circle. ℓ_{p}(Γ) L_{p}(μ) when μ is counting measure on the set Γ. ℓ_{p} ℓ_{p}(Γ) when Γ = ℕ. ℓ_{p}^{n} ℓ_{p}(Γ) when Γ = {1, 2, …, n}. c The subspace of ℓ_{∞} of scalar sequences which have a limit. c_{0}(Γ) The closure in ℓ_{∞}(Γ) of the scalar sequences which have finite support. c_{0} c_{0}(Γ) when Γ = ℕ. L_{p,q}(μ, X) The μ-measurable X-valued functions f for which

∥f∥_{p,q} := (\frac{q}{p} ∫_{0}^{∞} ∥t^{1/p}f^{∗}(t)∥_{X}^{q} \frac{d}{t})^{1/q} < ∞ (section 11). Here 0 < p, q < ∞ and f^{∗} is the decreasing rearrangement of ∥f∥_{X}.

L_{p,∞}(μ, X) The μ-measurable X-valued functions f for which ∥f∥_{p,∞} := sup_{t>0} t^{1/p}∥f^{∗}(t)∥_{X} < ∞ (section 11). Here 0 < p ≤ ∞ and f^{∗} is the decreasing rearrangement of ∥f∥_{X}.

L_{p,q}(μ) L_{p,q}(μ, X) when X = ℝ. B_{X} The closed unit ball of the Banach space X. B_{X}(x, r) The closed ball of radius r with center x in the Banach space X;

denoted also B(x, r) when X is understood. S^{⊥} All linear functionals which vanish on S [when S is a subset of a Banach

space]. S_{⊥} The intersection of the kernels of all linear functionals in S [when S is a

subset of the dual of a Banach space]. d(X, Y) The Banach-Mazur distance from X to Y (section 2). X ≈ Y The space X is isomorphic to the space Y. δ_{X}(·) The modulus of convexity of the space X (section 6). ρ_{X}(·) The modulus of smoothness of the space X (section 6). X_{u} When X is a lattice and u ≥ 0, the abstract M-space which has the

order interval [−u, u] as the unit ball (section 5). X_{u^{∗}} When X is a lattice and u^{∗} ≥ 0 in X^{∗}, the abstract L_{1}-space which is

the completion of X under the seminorm ∥x∥_{u^{∗}} = u^{∗}(|x|) (section 5). {ε_{n}}_{n=1}^{∞} A Rademacher sequence (section 4). I_{X} The identity operator on the space X. J_{X} The canonical embedding of X into X^{∗∗}. I_{p} The formal identity operator from C(K) to L_{p}(μ) (when μ is a finite

measure on the compact Hausdorff space K). I_{p,q} The formal identity mapping from L_{p}(μ) to L_{q}(μ).

B(X, Y) The bounded operators from X to Y. K(X, Y) The compact operators from X to Y. W K(X, Y) The weakly compact operators from X to Y. SS(X, Y) The strictly singular operators from X to Y (section 10). F r(X, Y) The Fredholm operators from X to Y (section 10). N (X, Y) The nuclear operators from X to Y (section 8). N (T) The nuclear norm of the operator T (section 8). M^{(p)}(T) The p-convexity constant of the operator T (section 5). M_{(p)}(T) The p-concavity constant of the operator T (section 5). T_{p}(X) The type p constant of the Banach space X (section 8). C_{p}(X) The cotype p constant of the Banach space X (section 8). Π_{p}(X, Y) The p-summing operators from X to Y (section 10). π_{p}(T) The p-summing norm of the operator T (section 10). I_{p}(X, Y) The p-integral operators from X to Y (section 10). i_{p}(T) The p-integral norm of the operator T (section 10).

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