Subspaces of L_{p} Isometric to Subspaces of ℓ_{p}

F. Delbaen , H. Jarchow^{(1)}, A. Pełczyński^{(2)}

Abstract. We present three results on isometric embeddings of a (closed, linear) subspace X of L_{p} = L_{p} 0, 1 into ℓ_{p}. First we show that if p ∉ 2ℕ, then X is isometrically isomorphic to a subspace of ℓ_{p} if and only if some, equivalently every, subspace of L_{p} which contains the constant functions and which is isometrically isomorphic to X, consists of functions having discrete distribution. In contrast, if p ∈ 2ℕ and X is finite - dimensional, then X is isometrically isomorphic to a subspace of ℓ_{p}^{N}, where the positive integer N depends on the dimension of X, on p, and on the chosen scalar field. The third result, stated in local terms, shows in particular that if p is not an even integer, then no finite - dimensional Banach space can be isometrically universal for the 2 - dimensional subspaces of L_{p}.

0. Introduction

Statement of Results

The starting point of the present paper was a question of Albrecht Pietsch: “Which finite - dimensional subspaces of L_{p} are isometrically isomorphic to subspaces of ℓ_{p}?” The purpose of this note is to give a comprehensive answer to Pietsch’s question. Curiously, the answer depends on the arithmetic nature of p and splits into two cases. In the first case p is an arbitrary positive number but not an even integer, in the second case p is a positive even integer (in symbols p ∈ 2ℕ). In the first case, our criterion extends to arbitrary, not necessarily finite - dimensional, closed linear subspaces of L_{p}. In any case, the answer is independent of whether the basic scalar field, which we denote by 𝕂, is the real number field ℝ or the complex number field ℂ.

Accordingly, the answer is stated in two parts, Theorem A and Theorem B, which are proved in Sections 1 and 2, respectively. The first theorem reads as follows:

Theorem A. Let 0 < p < ∞, p ∉ 2ℕ. Then a closed linear subspace X of L_{p} is isometrically isomorphic to a subspace of ℓ_{p} if and only if every, equivalently some, unital subspace which is isometrically isomorphic to X, consists of functions with discrete distribution.

1991 Mathematics Subject Classification: Primary 46 B 04, 46 E 30; Secondary 46 B 25, 46 B 07 . Key words and phrases. Subspaces of L_{p} and ℓ_{p}, linear isometries, distribution measures, equimeasurable functions, Banach-Mazur compactum, isometrically universal finite - dimensional Banach spaces

(1) Supported by the Swiss National Funds.

(2) Supported by KBN grant 2P0 3A 03614.

Recall that a subspace of L_{p} is unital if it contains the constant functions. As we shall see (Lemma 1.2), every non - zero closed linear subspace of L_{p} is in fact isometrically isomorphic to a unital one.

¿From Theorem A we can easily derive that if 0 < p < ∞ and p ∉ 2ℕ, then 2 - dimensional Hilbert space is not isometrically isomorphic to a subspace of ℓ_{p} (cf. Corollary 1.8). A weaker result, with ℓ_{p} replaced by ℓ_{p}^{n} (n ∈ ℕ), was discovered by Yu. Lyubich Ly .

Theorem B. Let p ∈ 2ℕ. Then every finite - dimensional subspace of L_{p} is isometrically isomorphic to a subspace of ℓ_{p}.

Moreover, for each n ∈ ℕ there exists an N ∈ ℕ, depending on n, p and 𝕂, such that every n - dimensional subspace of L_{p} is isometrically isomorphic to a subspace of ℓ_{p}^{N}.

Denote by N(n, p, 𝕂) the least N with the property described in the ‘moreover part’ of Theorem B, and by H(n, p, 𝕂) the smallest of all integers H such that n - dimensional Hilbert space is isometrically isomorphic to a subspace of ℓ_{p}^{H}. In recent years, the numbers H(n, p, 𝕂) have been object of intensive study (see the survey Kö and the references there). The second part of Section 2 is devoted to a discussion of the relationship between the numbers H(n, p, 𝕂) and N(n, p, 𝕂). In particular, we show that H(2, 4, 𝕂) = N(2, 4, 𝕂) for both, 𝕂 = ℝ and 𝕂 = ℂ. As a byproduct we obtain the surprising fact that every 2 - dimensional subspace of real L_{4} has a symmetric basis of symmetric norm one.

To describe the topic of Section 3 recall that a Banach space Y is isometrically universal for a given family F of Banach spaces provided every member of F is isometrically isomorphic to a subspace of Y. In this language, the ‘moreover part’ of Theorem B can be restated as follows: if p ∈ 2ℕ then given any n ∈ ℕ, there is an N ∈ ℕ such that ℓ_{p}^{N} is isometrically universal for the family of all n - dimensional subspaces of L_{p}.

This is in marked contrast to the strongly ‘negative’ character of

Theorem C. Let 1 ≤ p < ∞, p ∉ 2ℕ, and let X be a Banach space which is isomorphic either to L_{p} or to ℓ_{p}. Then, for each k = 2, 3, …, there is no finite - dimensional Banach space which is isometrically universal for all k - dimensional subspaces of X.

Theorem C will be obtained as a simple consequence of Theorem 3.2 which is the main result of Section 3. It is stated in terms of ‘Local Theory of Banach Spaces’ and connects, for a given Banach space X, the problem of non-existence of finite - dimensional Banach spaces which are isometrically universal for all 2 - dimensional subspaces of X to ‘type and cotype’ of X.

Tools and Notation

Let 0 < p < ∞, and let μ be a non - negative measure defined on a σ - field Σ of subsets of a set Ω. By L_{p}(μ) = L_{p}(Ω, Σ, μ) we denote the usual quasi - Banach space of μ - equivalence classes of scalar valued functions on Ω which are p - absolutely integrable against μ. The quasi - norm on L_{p}(μ) (which is a norm when p ≥ 1) will be denoted by ∥ · ∥_{p}, or by ∥ · ∥_{L_{p}(μ)} if necessary; so

∥f∥_{p} = ( ∫_{Ω}|f|^{p}dμ)^{1/p} for f ∈ L_{p}(μ) .

If ν is a probability (=normalized) measure on a σ - field of subsets of a set Ω and f : Ω → 𝕂 is a ν - measurable function (=random variable), then the distribution of f is the probability measure on B(𝕂) denoted by μ_{f} an defined by

μ_{f}(A) = ν(f^{−1}(A))  for  A ∈ B(𝕂) .

Here we denote, for a topological space X, by B(X) the σ - field of all Borel subsets of X. The distribution is said to be discrete if it is of the form

μ_{f} = ∑_{j∈ℕ} a_{j}δ_{x_{j}} ,

where the x_{j}’s and a_{j}’s are scalars, with a_{j} ≥ 0 for all j and ∑_{j} a_{j} = 1; by δ_{x} we denote the measure of total mass 1 concentrated at x.

We are going to consider the following standard L_{p} spaces:

• L_{p} := L_{p}( 0, 1 , B, λ); here B = B( 0, 1 ), and λ is the usual Lebesgue probability measure on B.

• ℓ_{p} is the space of all infinite p - absolutely summable sequences; so Ω is ℕ, Σ is the collection of all subsets of ℕ, and μ is the usual counting measure.

• Given N ∈ ℕ, the spaces ℓ_{p}^{N} and L_{p}^{N} are obtained by taking, for any N - point set M, μ to be the counting measure, resp. the normalized counting measure, on the subsets of M. The main tool to prove Theorem A is the following

Equimeasurability Lemma 0.1. Suppose that 0 < p < ∞, p ∉ 2ℕ, and that μ and ν are probability measures. If f ∈ L_{p}(μ) and g ∈ L_{p}(ν) are such that

∥1 + z·f∥_{p} = ∥1 + z·g∥_{p} for all z ∈ 𝕂 ,

then μ_{f} = μ_{g}.

The Equimeasurability Lemma will also enable us to prove Lemma 3.3, and this will eventually lead to a proof of Theorem C.

The lemma is a special case of a more general result discovered independently by A.I. Plotkin Pl 1 , Pl 2 , Pl 3 and by W. Rudin Ru ; cf. also Har , Theorem 1.1. For further information we refer to Ko and the references given there.

Our proof of Theorem B is based upon the classical theorem by C. Carathéodory on convex combinations in ℝ^{n} (see e.g. DGK ). The method goes back to J.H.B. Kemperman’s Ke research in empirical statistics; in fact, his result can be adopted to get the qualitative version of Theorem B.

Our strategy to prove Theorem C has its origin in a paper of C. Bessaga Be . Solving a problem of S. Mazur SB , he proved that there is no finite - dimensional Banach space which is isometrically universal for all 2 - dimensional Banach spaces. Hence, given any integer k ≥ 2, no finite - dimensional Banach space can be isometrically universal for all k - dimensional Banach spaces. Bessaga’s method was considerably extended by V.L. Klee, Jr. Kl , and in fact our proof of Theorem 3.2 is based upon Klee’s criterion on estimates of the topological dimension of families of finite - dimensional Banach spaces amitting finite - dimensional universal spaces (cf. Kl , §3). We will outline Klee’s argument in the Appendix to Section 3.

In Sections 2 and 3, we use the concept of Banach - Mazur compactum. Recall that given n ∈ ℕ and n - dimensional Banach space E and F, the (multiplicative) Banach - Mazur distance is defined by

d(E, F) := inf {c : ∃ u : E → F such that ∥x∥ ≤ ∥u(x)∥ ≤ c · ∥x∥ ∀ x ∈ E} .

The Banach - Mazur ‘metric’

δ(E, F) := log d(E, F)

induces a genuine metric (also denoted by δ( · , ·)) on

BM_{n} := the set of isometry classes of n - dimensional Banach spaces.

With respect to this metric BM_{n} is compact; it is called the n - th Banach - Mazur compactum.

In the sequel, we shall identify an element in BM_{n} with any of its representing n - dimensional Banach spaces. Accordingly, given a Banach space X, we define

BM_{n}(X) := {E ⊂ X : E ∈ BM_{n}} .

The following is well - known and easy to prove:

Lemma 0.2. If X is finite - dimensional then BM_{n}(X) is compact; more precisely, BM_{n}(X) is compact when it is identified with a subset of the space (BM_{n}, δ).

Throughout this paper, terms like ‘isomorphism’, ‘isomorphic embedding’, ‘isometric isomorphism’, … have their usual meaning; see DS . Frequently, we shall use ‘subspace’ instead of ‘closed linear subspace’, ‘operator’ instead of ‘bounded linear map’, etc., and occasionally even ‘isometric’ instead of ‘isometrically isomorphic’.

We employ standard set theoretic notation. The number of elements in a set Ω is denoted by |Ω| (where |Ω| = ∞ if Ω is infinite). We write 2^{Ω} for the family of all subsets of Ω, and we denote the indicator function of a subset C of Ω by 1_{C}. Instead of 1_{Ω}, we simply write 1.

The support of a function f : Ω → 𝕂 is the set {ω ∈ Ω: f(ω) ≠ 0}. We adopt the convention to understand equalities and inclusions involving measurable sets modulo null sets. Accordingly, if f is an equivalence class of measurable functions Ω → 𝕂, we denote by supp f the support of any representative of f. Let X be a space of such (equivalence classes of) functions. We say that f ∈ X has full support in X if μ(supp g \ supp f) = 0 for any g ∈ X. Clearly, if μ is a probability measure, then f ∈ L_{p}(μ) has full support (in L_{p}(μ)) iff μ(supp f) = 1.

It should be clear how to extend the preceding results to spaces L_{p}(μ) where μ is an arbitrary infinite, σ - finite, separable measure.

Acknowledgement

We are particularly indepted to Stanisław Kwapień for his constant interest in our research. He proved Theorem 1.3 and kindly permitted us to publish the result. He also suggested the use of Auerbach bases in the study of 2 - dimensional subspaces of L_{4}. Further, we are indebted to Stanisław J. Szarek who is a coauthor of the proofs of Propositions 2.1 and 2.4. Finally, our thanks go to Czesław Bessaga for his constructive critics of preliminary versions of Section 3 and the Appendix. Most of this research has beem done while the third named author was visiting ETH Zürich and the University of Zürich in October 1997.He would like to thanks these institutions for their support and excellent working conditions.

1. Proof of Theorem A

We begin with two known lemmas. The first one was essentially established by S. Banach (cf. Chapter IX in Ba ). In a general setting, it is due to J. Lamperti La . We present here a simple proof of the original result based on a formula taken from Ro , p. 48.

Lemma 1.1. Let 0 < p < ∞ and suppose that f ∈ L_{p} satisfies (i) ∥f∥_{p} = 1 and (ii) λ(supp f) = 1. Then there exists an isometric isomorphism of L_{p} onto itself, say v, such that vf = 1 and |vg| ≤ 1 for all g ∈ L_{p} with |g| ≤ |f|.

Proof. Let us put Φ(t) := ∫_{0}^{t} |f(s)|^{p} ds for t ∈ 0, 1 . It follows from (i) and (ii) that Φ is an absolutely continuous, strictly increasing homeomorphism of 0, 1 onto itself. Moreover, \frac{dΦ}{dt} = |f|^{p} λ - a.e. By (i), μ := |f|^{p}dλ is a probability measure on B.

Clearly, if 0 ≤ a < b ≤ 1 then

μ( a, b ) = ∫_{a}^{b}|f(t)|^{p}dt = Φ(b) − Φ(a) = λ(Φ( a, b )) .

The measure μ is obviously regular. Since Φ is absolutely continuous, this is also true for λ ∘ Φ. Regular measures on B are uniquely determined by their values on closed subintervals, hence

μ(A) = λ(Φ(A)) for every B ∈ B .

Let Ψ : 0, 1 → 0, 1 be the inverse of Φ. Since this is a homeomorphism, the map u : L_{p}(μ) → L_{p} defined by u(h) = h ∘ Ψ is an isometric isomorphism onto.

On the other hand, it follows from (ii) that w(g) := g/f also defines an isometric isomorphism w of L_{p} onto itself. Thus v = u ∘ w maps L_{p} isometrically onto itself. Obviously,

vg = \frac{g ∘ Ψ}{f ∘ Ψ} for g ∈ L_{p} ,

so that v has the desired properties.

Lemma 1.2. Let 0 < p < ∞. Then every non - zero subspace X of L_{p} is isometrically isomorphic to a unital subspace of L_{p}.

Proof. Together with L_{p}, X is separable. Hence, by Har , Lemma 3.2, there exists a g ∈ X with full support in X. Note that λ(A) > 0 since X ≠ {0}. Consider the probability space (A, B_{A}, λ_{A}) where B_{A} := {B ∈ B : B ⊂ A} and λ_{A} is the conditional probability defined by λ_{A}(B) = λ(B) · λ(A)^{−1} for B ∈ B_{A}. Note that f ↦ λ(A)^{−1/p}f defines an isometric isomorphism of X onto a subspace of L_{p}(λ_{A}) and that λ_{A} is a purely nonatomic separable probability measure.

It follows from Hal , Chap. VIII, Theorem 3, that the measure algebras (A, B_{A}, λ_{A}) and ( 0, 1 , B, λ) are isometrically isomorphic. The isometric isomorphism of the algebras induces an isometric isomorphism, say u, of L_{p}(λ_{A}) onto L_{p} such that λ_{A}(supp g) = λ(supp u(g)). In particular, λ(supp u(g)) = 1, and Lemma 1.1 provides us with an isometric isomorphism v of L_{p} onto itself such that v(u(g)) = 1. Hence X is isometrically isomorphic to the unital subspace vu(X) of L_{p}.

Proof of Theorem A. Necessity:

Let p ∉ 2ℕ. Let X be a unital subspace of L_{p} which is isometrically isomorphic to a subspace of ℓ_{p}, via an isometric isomorphism w.

By Har , Lemma 3.4, γ := w(1) has full support in w(X). Put S := supp γ. Clearly, w(X) can be regarded as a subspace of ℓ_{p}^{S}, where ℓ_{p}^{S} = {ξ ∈ ℓ_{p} : ξ(s) = 0 for s ∉ S}. (S is not necessarily a finite set!) Define the discrete measure ν on 2^{S} by

ν(B) := ∑_{s∈B} |γ(s)|^{p} for B ⊂ S .

Note that ν is a probability measure because ν(S) = ∑_{s∈S} |γ(s)|^{p} = ∥w(1)∥_{ℓ_{p}}^{p} = 1. Next define u : ℓ_{p}^{S} → L_{p}(ν) by

u(ξ) := (\frac{ξ(s)}{γ(s)} · sign γ(s))_{s∈S} (ξ ∈ ℓ_{p}^{S}) .

This is an isometric isomorphism with u(γ) = 1. Hence uw is an isometric isomorphism of X onto the unital subspace uw(X) of L_{p}(ν) such that uw(1) = 1.

Now pick f ∈ X. Clearly, if f = z · 1 for some z ∈ 𝕂, then f has discrete distribution. If f ≠ z · 1 for all z ∈ 𝕂, then uw carries the 2 - dimensional subspace of X spanned by 1 and f onto the 2 - dimensional subspace of L_{p}(ν) spanned by 1 and uw(f). Recall that p ∉ 2ℕ and invoke the Equimeasurability Lemma 0.1 to conclude that uw(f) has discrete distribution since f does.

Proof of Theorem A. Sufficiency:

Let X be a (unital) subspace of L_{p} such that each f ∈ X has discrete distribution. Under the additional hypothesis that X is finite - dimensional the proof is easy.

In fact, let then {f_{1}, …, f_{k}} be a basis for X. Since each f_{n} (1 ≤ n ≤ k) has discrete distribution, we may write

f_{n} = ∑_{j∈J(n)} z_{j,n}1_{C(j,n)}

where either J(n) = ℕ, or J(n) is a finite interval of ℕ, the z_{j,n}’s belong to 𝕂 (j ∈ J(n)) and (C(j, n))_{j∈J(n)} is a partition of 0, 1 into Borel sets. Changing, if necessary, each C(j, n) by an appropriate λ - null set, we may also assume that for every sequence (j(n))_{n=1}^{k} ∈ ∏_{n=1}^{k} J(n) we have

⋂_{n=1}^{k} C(j(n), n) = ∅  whenever  λ( ⋂_{n=1}^{k} C(j(n), n)) = 0 .

Let A be the Boolean algebra generated by the sets C(j(n), n), j(n) ∈ J(n), n = 1, …, k. It is easy to verify that A is purely atomic; the atoms being all the sets ⋂_{n=1}^{k}C(j(n), n) of positive measure. Thus, if Y is the closed subspace of L_{p} generated by the characteristic functions of these atoms, then Y is isometrically isomorphic either to ℓ_{p}, or to ℓ_{p}^{m} for some m ∈ ℕ. Clearly X ⊂ Y because f_{n} ∈ Y for n = 1, …, k.

In the general case the proof of the sufficiency part of Theorem A is much deeper. It is an immediate consequence of Theorem 1.3 below which was conjectured by the authors and essentially proved by S. Kwapień. His original elegant argument was more probabilistic in nature; it made extensive use of Prokhorov’s Theorem and the concept of weak convergence of distribution measures (cf. Pa ). We present here an altenative argument.

Theorem 1.3. Let 0 < p < ∞ and let X be a subspace of L_{p} such that each function in X has discrete distribution. Then X is isometrically isomorphic to a subspace of ℓ_{p}.

The proof requires a bit of preparation.

We work on a fixed probability space (Ω, Σ, μ). Recall that (Ω, Σ, μ) can be decomposed into at most countably many sets A_{n}, A_{∞} ∈ Σ such that each A_{n} is an atom whereas A_{∞} contains no atoms. If μ(A_{∞}) = 1, then (Ω, Σ, μ) is called atomless or diffuse. If μ(A_{∞}) = 0, then (Ω, Σ, μ) is called purely atomic. Suppose that μ(A_{∞}) > 0. If g : Ω → 𝕂 is essentially injective in the sense that {(ω, ω′) ∈ Ω×Ω : ω ≠ ω′, g(ω) = g(ω′)} is a null set for the product measure μ ⊗ μ, then g cannot have atomic distribution. Indeed, it suffices to remark that A_{∞} is then essentially uncountable.

(The problem whether A_{∞} has at least the power of the continuum is fundamental in set theory and leads to discussions on measurable cardinals, fortunately we do not need such results here.)

Given a set M of measurable functions Ω → 𝕂, we denote the σ - algebra generated by the members of M by σ(M), or by a related simpler symbol. We need a lemma on pairs of measurable functions g, h : Ω → 𝕂. It is clear that g has atomic distribution if and only if σ(g) is generated by a partition of Ω into at most countably many atoms. If the functions g, h : Ω → 𝕂 have atomic distribution and α ∈ 𝕂, then g + αh also has atomic distribution, and the partition generating σ(g + αh) is coarser than σ(g, h).

Lemma 1.4. Suppose that g and h have atomic distribution. Then there is, for each ε > 0, an α ∈ 𝕂 such that |α| ≤ ε and

σ(g + αh) = σ(g, h) .

Proof. Let (A_{s}) and (B_{t}) be the (at most countable) partitions of Ω generating σ(g) and σ(h), respectively. Then g = ∑_{s} ξ_{s}1_{A_{s}} and h = ∑_{t} η_{t}1_{B_{t}} where ξ_{s} ≠ ξ_{s′} if s ≠ s′ and η_{t} ≠ η_{t′} if t ≠ t′. Consider the set Z of all (ξ_{s} − ξ_{s′})/(η_{t} − η_{t′}) with t ≠ t′. Since Z is at most countable, we may select α ∈ 𝕂 such that |α| ≤ ε and α ∉ Z. We claim that σ(g + αh) = σ(g, h).

For this it is enough to show that if ω ∈ A_{s} ∩ B_{t} and ω′ ∈ A_{s′} ∩ B_{t′} are such that (g + αh)(ω) = (g + αh)(ω′), then s = s′ and t = t′. Note that t = t′ yields ξ_{s} = ξ_{s′} and so s = s′ by the very definition. On the other hand, t ≠ t′ cannot happen! In fact, together with ξ_{s} + αη_{t} = ξ_{s′} + αη_{t′} this would lead to α = (ξ_{s} − ξ_{s′})/(η_{t′} − η_{t}) ∈ Z: a contradiction.

We denote by L_{0}(μ) the space of all (equivalence classes of) measurable functions Ω → 𝕂. Recall that with respect to convergence in measure, this is a complete metrizable topological vector space.

We are going to consider subsets M of L_{0}(μ) with ¿the following property:

For every sequence (f_{n}) in M there is a sequence (ε_{n}) of positive real numbers such (*) that whenever (a_{n}) is a sequence of scalars such that |a_{n}| ≤ ε_{n} ∀ n ∈ ℕ, then ∑_{n} a_{n}f_{n}

converges in measure and its limit belongs to M.

This condition is satisfied if M is a closed subspace of L_{p}(μ) for 0 ≤ p < ∞, or a weak∗ closed subspace of L_{∞}(μ). But M could also just be a sufficiently big generator of a suitable space of functions on Ω. Thus Theorem 1.3 is a special case of the following

Theorem 1.5. If M ⊂ L_{0}(μ) satisfies (∗) and every g ∈ M has atomic distribution, then there is an at most countable partition of Ω into sets A_{k} ∈ Σ such that for each f ∈ M there are scalars a_{k}(f) with

f = ∑_{k} a_{k}(f)1_{A_{k}}

(convergence in L_{0}(μ)).

Proof. We start with the case where M is a separable subset of L_{0}(μ). Let (f_{n})_{n∈ℕ} be a dense sequence in M. We may assume that Σ coincides with σ(M) = σ(f_{n} : n ∈ ℕ). In fact, otherwise a standard reduction procedure could be applied to arrange for such a situation. We may also assume that (f_{n}) separates the points of Ω in the sense that if ω, ω′ ∈ Ω are such that f_{n}(ω) = f_{n}(ω′) for all n, then ω = ω′. Again, should this not a priori be the case, we could pass from Ω to an appropriate set of equivalence classes of points in Ω.

All what is then needed to show is that σ({f_{n} : n}) is purely atomic. The idea is to find scalars a_{n} such that g = ∑_{n} a_{n}f_{n} exists in M and is essentially injective in the sense explained above. Since g has atomic distribution, it is clear that there cannot be any diffuse part in (Ω, Σ, μ).

The construction is as follows. Fix (ε_{n}) as in (∗). The trick is to select a_{n}’s with |a_{n}| ≤ ε_{n} so that (a_{n}) tends to zero so quickly that g = ∑_{n} a_{n}f_{n} must be injective. We proceed by induction.

For n = 1 we simply take a_{1} = ε_{1} (which we can take to be 1 if we like). Suppose that, for some n, we have already found a_{1}, …, a_{n} such that σ(∑_{k=1}^{n} a_{k}f_{k}) = σ(f_{1}, …, f_{n}). Let (D_{j}^{(n)})_{j} be the (at most countable) family of atoms of σ(f_{1}, …, f_{n}). Choose N_{n} ∈ ℕ so big that μ( ⋃_{j>N_{n}} D_{j}^{(n)}) ≤ 3^{−n}. On ⋃_{j≤N_{n}} D_{j}^{(n)}, the function a_{1}f_{1} + … + a_{n}f_{n} assumes N_{n} distinct values ξ_{1}, …, ξ_{N_{n}}. Choose 0 < δ_{n} ≤ δ_{n−1} such that δ_{n} ≤ |ξ_{j} − ξ_{j′}| for all j, j′ ≤ N_{n} with j ≠ j′ and fix m_{n+1} so that μ{|f_{n+1}| > m_{n+1}} ≤ 3^{−n}. Then we can find a_{n+1} such that

(i) |a_{n+1}| ≤ ε_{n+1} , (ii) |a_{n+1}| ≤ m_{n+1}^{−1}3^{−n}δ_{n}, (iii) σ(a_{1}f_{1} + … + a_{n+1}f_{n+1}) = σ(a_{1}f_{1} + … + a_{n}f_{n}, f_{n+1}). In the last step we have applied Lemma 1.4. Note that the induction hypothesis allows us to write σ(a_{1}f_{1} + … + a_{n}f_{n}, f_{n+1}) = σ(f_{1}, …, f_{n}, f_{n+1}).

Let now g = ∑_{n} a_{n}f_{n}. Because of Σ = σ(f_{n} : n ∈ ℕ), {(ω, ω′) : g(ω) = g(ω′)} is an element of Σ ⊗ Σ. For each n, define now E_{n} ∈ Σ ⊗ Σ by

E_{n} := (( ⋃_{j≤N_{n}} D_{j}^{(n)})×( ⋃_{j≤N_{n}} D_{j}^{(n)}))^{c} ∪ ⋃_{k>n}{(ω, ω′) : |f_{k}(ω)| > m_{k} or |f_{k}(ω′)| > m_{k}} .

Here A^{c} stands for the complement of a set A. A straightforward calculation reveals that

(μ ⊗ μ)(E_{n}) ≤ 3^{−n} + 3^{−n} + ∑_{k>n}(3^{−k} + 3^{−k}) = 3^{−n+1} .

The Borel - Cantelli Lemma (cf. Kh , p.7) now implies that for μ ⊗ μ almost all choices of (ω, ω′) ∈ Ω×Ω, there is an n_{0} = n_{0}(ω, ω′) such that (ω, ω′) ∈ E_{n}^{c} for all n ≥ n_{0}.

Suppose now that in addition ω, ω′ are such that g(ω) = g(ω′) and let n ≥ n_{0}. Clearly, ω, ω′ both belong to ⋃_{j≤N_{n}} D_{j}^{(n)} and max {|f_{k}(ω)|, |f_{k}(ω′)|} ≤ m_{k} whenever k > n. Our assumptions on (a_{k})_{k} imply that

| ∑_{k>n} a_{k}f_{k}(ω)| ≤ ∑_{k>n} |a_{k}||f_{k}(ω)| ≤ ∑_{k>n} 3^{−k}δ_{k} ≤ δ_{n} ∑_{k>n} 3^{−k} = 3^{−n}δ_{n}/2 .

Similar for ω′. Thus taking into account that g(ω) = g(ω′) we get

| ∑_{k≤n} a_{k}f_{k}(ω) − ∑_{k≤n} a_{k}f_{k}(ω′)| = | ∑_{k>n} a_{k}f_{k}(ω′) − ∑_{k>n} a_{k}f_{k}(ω)| ≤ 3^{−n}δ_{n} ,

hence ω, ω′ are in the same atom D_{j}^{(n)} for some j ≤ N_{n}. Thus ∑_{k≤n} a_{k}f_{k}(ω) = ∑_{k≤n} a_{k}f_{k}(ω′) and so, by our choice of the sequence (a_{s})_{s}, we have f_{k}(ω) = f_{k}(ω′) for k ≤ n. But n ≥ n_{0} was arbitrary, hence f_{k}(ω) = f_{k}(ω′) for all k.

This leads to ω = ω′ and completes the proof for the case when M is separable, viz. σ(X) is generated by a sequence of functions. The general case can be reduced to this one as follows.

Let S := {Σ_{i} : i ∈ I} be the collection of all σ - algebras which are generated by sequences in M. S is a σ - lattice in the sense that for each countable family (Σ_{i_{n}})_{n} in S there is a j ∈ I such that Σ_{i_{n}} ⊂ Σ_{j} for all n.

The preceding reasoning applied to the set M_{i} := {f ∈ M : f is Σ_{i} - measurable} shows that each Σ_{i} is purely atomic. For each i ∈ I, we set φ(i) := ∑_{D} μ(D)^{2} where summation is over all atoms D of Σ_{i}. It is clear that φ(i) ≥ φ(j) whenever Σ_{i} ⊂ Σ_{j}, with equality only when Σ_{i} = Σ_{j} (up to sets of measure zero). Since S is a σ - lattice and since φ(i) > 0 for each i, we get that φ attains its minimum: φ(j) = min {φ(i) : i ∈ I} for some j ∈ I. It follows that Σ_{j} is the biggest of all our σ-algebras, that is, that Σ_{j} = σ(M).

The proof is complete.

Corollary 1.6. Suppose that X is a Banach space of measurable functions on Ω, that X ↪ L_{0}(μ) is continuous, and that M ⊂ X is a closed absolutely convex subset whose linear span is dense in X for the L_{0}(μ) topology. If all functions in M have purely atomic distribution, then the same is true for all functions in X; there is even a partition of Ω into at most countably many atoms A_{n} such that each f ∈ X has a representation f = ∑_{k} a_{k}(f)1_{A_{k}}, the a_{k}(f)’s being suitably chosen scalars.

The next corollary is an immediate consequence of Theorem A.

Corollary 1.7. If 0 < p < ∞ is not an even integer, then a subspace X of L_{p} is isometrically isomorphic to a subspace of ℓ_{p} if and only if every 2 - dimensional subspace of X is isometrically isomorphic to a subspace of ℓ_{p}.

The last result of this section follows from Corollary 1.7 and ¿from the fact that if p ≠ 2, then no infinite - dimensional subspace of ℓ_{p} can be isomorphic to a Hilbert space.

Corollary 1.8. If 0 < p < ∞ is not an even integer, then the 2 - dimensional Hilbert space ℓ_{2}^{2} cannot be isometrically isomorphic to a subspace of ℓ_{p}.

It is interesting to compare Theorem A with Theorem 5.4 of N.J. Kalton and D. Werner KW (see also GK where the case p = 1 is studied). They proved that if 1 < p < ∞ and if X is a subspace of L_{p}, then the unit ball of X is compact in the norm topology of L_{1} if and only if X is ‘nearly isometric’ to a subspace of ℓ_{p} in the sense that, for each ε > 0, there exists an isomorphism u_{ε} : X → ℓ_{p} such that ∥x∥ ≤ ∥u_{ε}(x)∥ ≤ (1 + ε)∥x∥ for all x ∈ X. Clearly every finite - dimensional subspace of L_{p} has this property. An interesting infinite - dimensional example is the Müntz space M_{p}({t^{n_{k}}}); it is defined to be the subspace of L_{p} generated by the monomials t^{n_{k}}, k ∈ ℕ, where (n_{k}) is such that ∑_{k=1}^{∞} n_{k}^{−1} < ∞. Combining Theorem 5.4 of KW with results of Sch we infer that M_{p}({t^{n_{k}}}) is nearly isometric to a subspace of ℓ_{p} for 1 < p < ∞, whereas Theorem A yields that, at least for p ∉ 2ℕ, M_{p}({t^{n_{k}}}) is not isometrically isomorphic to a subspace of ℓ_{p}.

2. Proof of Theorem B

In this section, p will be an even integer: p = 2s with s ∈ ℕ.

Proof of Theorem B. It suffices to show that there exist an integer N, depending on n, p and 𝕂, and a subset A_{n} of BM_{n}(L_{p}), dense with respect to the Banach - Mazur metric, such that A_{n} ⊂ BM_{n}(ℓ_{p}^{N}). Continuity of the embedding and compactness of BM_{n}(ℓ_{p}^{N}) will then yield BM_{n}(L_{p}) ⊂ BM_{n}(ℓ_{p}^{N}).

We choose A_{n} as follows. Fix r ∈ ℕ and use the intervals I_{k,r} := [\frac{k−1}{r}, \frac{k}{r} ], k = 1, …, r, to define the finite rank operator u_{r} : L_{p} → L_{p} by

u_{r}(f) := ∑_{k=1}^{r} r · ( ∫_{0}^{1}f(t) · 1_{I_{k,r}}(t) dt) · 1_{I_{k,r}} (f ∈ L_{p}) .

Let A_{n,r} ⊂ BM_{n}(L_{p}) be the family generated by all n - dimensional subspaces of u_{r}(L_{p}), and put

A_{n} := ⋃_{r∈ℕ} A_{n,r} .

We claim that A_{n} is dense in BM_{n}(L_{p}). Indeed, it is well - known and easy to check that the operators u_{r} are norm - one projections such that lim_{r} ∥u_{r}(1_{[a,b]}) − 1_{[a,b]}∥_{p} = 0 for every interval [a, b] ⊂ [0, 1]. It follows that the sequence (u_{r}) converges strongly to the identity operator on L_{p}. By the Banach - Steinhaus principle, it tends to the identity uniformly on compact sets. In particular, if F is an n - dimensional subspace of L_{p} then, given ε with 0 < ε < 1, we get

(1 − ε)∥f∥_{p} ≤ ∥u_{r}(f)∥_{p} ≤ ∥f∥_{p}

for r sufficiently large. Hence d(F, u_{r}(F)) ≤ (1 − ε)^{−1}. This proves the claim.

Clearly, u_{r}(1) = 1, so that if F is unital then so is u_{r}(F). In order to complete the proof of Theorem B, it is therefore enough to show that if E is a unital n - dimensional subspace of some u_{r}(L_{p}), then E is isometrically isomorphic to a subspace of ℓ_{p}^{N} for some N independent of E and r.

We present the argument in the case 𝕂 = ℂ.

Given n ∈ ℕ, we choose N = N_{ℂ} to be the number of elements of the subset S = S(n, p, ℂ) of ℤ_{+}^{n−1}×ℤ_{+}^{n−1} \ {(0, 0)} defined by

S := {(α, β) ∈ ℤ_{+}^{n−1}×ℤ_{+}^{n−1} : ∑_{j=1}^{n−1} α_{j} ≤ s and ∑_{j=1}^{n−1} β_{j} ≤ s} .

Here ℤ_{+} := ℕ ∪ {0} and (0, 0) denotes the sequence in ℤ_{+}^{n−1}×ℤ_{+}^{n−1} with zero coordinates only.

In the sequel, we shall apply standard multiindex notation. So, if α = (α_{j})_{j=1}^{n−1} ∈ ℤ_{+}^{n−1} and z = (z_{j})_{j=1}^{n−1} ∈ ℂ^{n−1}, then z^{α} := ∏_{j=0}^{n−1} z_{j}^{α_{j}}. Here 0^{0} is understood to be 1. We also set \overline{z} := (\overline{z_{j}})_{j=1}^{n−1}, Re z := (Re z_{j})_{j=1}^{n−1} and Im z := (Im z_{j})_{j=1}^{n−1}, and we use corresponding notation for (n − 1) - tuples of scalar valued functions f_{n} in the pointwise sense. For example, if we are given F = (f_{j})_{j=1}^{n−1} with ℂ - valued functions f_{j}, then we write (Re F)^{α} = ∏_{j=0}^{n−1}(Re f_{j})^{α_{j}} for α ∈ ℤ_{+}^{n−1}.

We use ‘ < ’ to denote lexicographic order in ℤ_{+}^{n−1}.

Since E ⊂ u_{r}(L_{p}) ⊂ L_{∞} and since E is unital, E admits a basis {f_{0}, f_{1}, …, f_{n−1}} with f_{0} = 1 and ∥f_{j}∥_{∞} = 1 for j = 1, …, n − 1. Put F := (f_{j})_{j=1}^{n−1}. Then, for any choice of a_{0}, a_{1}, …, a_{n−1} ∈ ℂ,

\begin{aligned}∥ ∑_{j=0}^{n−1} a_{j}f_{j}∥_{p}^{p} & = ∫_{0}^{1}( ∑_{j=0}^{n−1} a_{j}f_{j})^{s}( ∑_{j=0}^{n−1} \overline{a_{j}f_{j}})^{s} dt \\ & = ∑_{α∈ℤ_{+}^{n−1}} c_{α,α} ∫_{0}^{1}F^{α}\overline{F}^{α} + ∑_{{(α,β)∈S \atop α≠β}}c_{α,β} ∫_{0}^{1}F^{α}\overline{F}^{β} dt , \end{aligned}

where the c_{α,β} are polynomials in the variables (a_{j})_{j=0}^{n−1} and (\overline{a_{j}})_{j=0}^{n−1} such that c_{α,β} = \overline{c_{β,α}}. Coupling the pairs (α, β) and (β, α) and using the identity

c_{α,β}F^{α}\overline{F}^{β} + c_{β,α}F^{β}\overline{F}^{α} = 2 Re c_{α,β}Re (F^{α}\overline{F}^{β}) − 2 Im c_{β,α}Im (F^{β}\overline{F}^{α})

we get

(2.1)  ∥ ∑_{j=0}^{n−1} a_{j}f_{j}∥_{p}^{p} = c_{0,0} + ∑_{(α,β)∈S} d_{α,β}m_{α,β}

where

(2.2)  m_{α,β} := \{ \begin{matrix}∫_{0}^{1}F^{α}\overline{F}^{α} dt & for α = β ≠ 0 \\ ∫_{0}^{1}Re (F^{α}\overline{F}^{β}) dt & for α < β \\ ∫_{0}^{1}Im (F^{β}\overline{F}^{α}) dt & for α > β \end{matrix}

and

d_{α,β} = \{ \begin{matrix}c_{α,α} & for α = β ≠ 0 \\ 2 Re c_{α,β} & for α < β \\ −2 Im c_{β,α} & for α > β . \end{matrix}

By construction,

m := (m_{α,β})_{(α,β)∈S}

is an element of ℝ^{N}.

Next we consider the manifold L ⊂ ℝ^{N} defined by

L = {w(z) = (w_{α,β}(z))_{(α,β)∈S} : z ∈ D^{n−1}} ,

where D is the closed unit disk {z ∈ ℂ : |z| ≤ 1} in the complex plane and

w_{α,β}(z) = \{ \begin{matrix}z^{α} \overline{z}^{α} & for α = β ≠ 0 \\ Re (z^{α} \overline{z}^{β}) & for α < β \\ Im (z^{β} \overline{z}^{α}) & for α > β . \end{matrix}

Note that the function z ↦ w(z) is a homeomorphism from D^{n−1} onto L and that, if δ^{(j)} = (δ_{j,k})_{k=1}^{n−1} is the j’th standard unit vector in ℤ_{+}^{(n−1)}, then

w_{0,δ^{(j)}}(z) − i · w_{δ^{(j)},0}(z) = z_{j}

for all z ∈ D^{n−1} and j = 1, …, n − 1.

We are going to work with the convex hull of L,

C := conv L .

We start by showing that

(2.3)  m ∈ C .

Indeed, since the f_{j}’s belong to u_{r}(L_{p}), they are constant on each of the intervals I_{k,r}, k = 1, …, r. It follows that the functions Re (F^{α}\overline{F}^{β}) and Im (F^{β}\overline{F}^{α}) do have the same property. We put

z^{(k)} := (f_{j}(\frac{k − \frac{1}{2}}{r} ))_{j=1}^{r} and w^{(k)} := w(z^{(k)}) (k = 1, …, r) .

¿From ∥f_{j}∥_{∞} = 1 for j = 1, …, n − 1 we infer that the w^{(k)}’s belong to L. Since m = \frac{1}{r} ∑_{k=1}^{r} w^{(k)}, the proof of (2.3) is complete.

Since L ⊂ ℝ^{N} is compact and connected, it follows ¿from an improvement of Carathéodory’s Theorem (see Fe and DGK , 3.5, p. 117) that m is a convex combination of at most N elements of L. Hence there are vectors ξ^{(1)}, …, ξ^{(N)} ∈ L and numbers ν_{1}, …, ν_{N} ≥ 0 such that ∑_{d=1}^{N} ν_{d} = 1 and

(2.4)  m = ∑_{d=1}^{N} ν_{d} ξ^{(d)} .

Let N := {1, …, N}. Define a probability measure ν on 2^{N} by

ν(A) := ∑_{d ∈A} ν_{d} (A ⊂ N) .

Let z^{(d)} = (z_{j}^{(d)})_{j=1}^{n−1} be the unique point of D^{n−1} such that ξ^{(d)} = w(z^{(d)}) for d ∈ N. Define functions v_{j} : N → ℂ by setting v_{0} := 1 and

v_{j}(d) := z_{j}^{(d)} for d ∈ N and j = 1, 2, …, n − 1 .

Let E_{0} be the subspace of L_{p}(ν) spanned by the functions v_{0}, …, v_{n−1}. We claim that E and E_{0} are isometrically isomorphic. In fact, we will show that the map f_{j} ↦ v_{j} (j = 0, …, n − 1) extends to an isometric isomorphism from E onto E_{0}.

To prove this, put V = (v_{j})_{j=1}^{n−1} and notice first that

(2.5)  m_{α,β} := \{ \begin{matrix}∫_{N} V^{α} \overline{V}^{α} dν & for α = β ≠ 0 \\ ∫_{N} Re (V^{α}\overline{V}^{β}) dν & for α < β \\ ∫_{N} Im (V^{β}\overline{V}^{α}) dν & for α > β . \end{matrix}

In fact, if α < β then, in view of (2.4),

\begin{aligned}∫_{N} Re (V^{α}\overline{V}^{β}) dν & = ∑_{d=1}^{N} ν_{d} · Re ( ∏_{j=0}^{n−1} v_{j}(d)^{α_{j}} \overline{v}_{j}(d)^{β_{j}}) = ∑_{d=1}^{N} ν_{d} · Re ((z^{(d)})^{α}(\overline{z^{(d)}})^{β}) \\ & = ∑_{d=1}^{N} ν_{d} w_{α,β}(z^{(d)}) = ∑_{d=1}^{N} ν_{d} ξ_{l,β}^{(d)} = m_{α,β} . \end{aligned}

Similarly for α = β and α > β.

Combining (2.1), (2.2) and (2.5) we get what we wanted: no matter how we choose complex numbers a_{0}, …, a_{n−1}, we have

∥ ∑_{j=0}^{n−1} a_{j}v_{j}∥_{L_{p}(ν)}^{p} = ∫_{N} ( ∑_{j=0}^{n−1} a_{j}v_{j})^{s}( ∑_{j=0}^{n−1} \overline{a_{j}v_{j}})^{s} dν

= ∑_{α∈ℤ_{+}^{n−1}} c_{α,α} ∫_{N} V^{α}\overline{V}^{α}dν + ∑_{{(α,β)∈S \atop α≠β}}c_{α,β} ∫_{N} V^{α}\overline{V}^{β}dν

= c_{0,0} + ∑_{(α,β)∈S} d_{α,β}m_{α,β} = ∥ ∑_{j=0}^{n−1} a_{j}f_{j}∥_{p}^{p} .

To complete the proof of Theorem B in the case 𝕂 = ℂ, just note that L_{p}(ν) embeds isometrically into ℓ_{p}^{N} via g ↦ (g(d)ν_{d}^{1/p})_{d=1}^{N}. (This map is not necessarily onto because some of the ν_{d}’s might be zero.)

The proof in the real case is similar but simpler. We define N = N_{ℝ} to be the number of elements of the set

S(n, p, ℝ) = {α ∈ ℤ_{+}^{n−1} : 1 ≤ ∑_{j=1}^{n−1} α_{j} ≤ p} .

Let now N(n, p, 𝕂) denote the smallest of all integers N such that every n - dimensional subspace of L_{p} is isometrically isomorphic to a subspace of ℓ_{p}^{N}. It follows from our construction and a well - known combinatorial argument giving the number of different distributions of p identical balls in n − 1 boxes that

N(n, p, ℝ) ≤ N_{ℝ} = ({n + p − 1 \atop p}) − 1 ,

(2.6)

N(n, p, ℂ) ≤ N_{ℂ} = ({n + s − 1 \atop s})^{2} − 1 .

Estimates for N(n, p, 𝕂) from below can possibly be obtained using results ¿from BLM on almost isometric embeddings of finite - dimensional subspaces of L_{p} into ℓ_{p}^{r}. An alternative approach is as follows.

Let H(n, p, 𝕂) denote the smallest of all integers H such that n - dimensional Hilbert space embeds isometrically into ℓ_{p}^{H} (all spaces are over 𝕂). The numbers H(n, p, 𝕂) have been objects of intensive study starting from Mi (cf. Mi , in particular p. 284, Problem). For the state of affairs we refer to Kö and references given there, and to the very recent note LS where the case n = 2 is discussed.

Obviously,

H(n, p, 𝕂) ≤ N(n, p, 𝕂) .

The upper estimates for H(n, p, 𝕂) given in Kö , Proposition 2, are essentially the same as ours for N(n, p, 𝕂) given in (2.6). This suggests the

Conjecture: H(n, p, 𝕂) = N(n, p, 𝕂) .

We arrived at this conjecture discussing the problem with J. Lindenstrauss.

The next two propositions show that H(2, 4, 𝕂) = N(2, 4, 𝕂). Their proofs have been obtained in collaboration with S. Kwapień and S. Szarek.

Proposition 2.1. Every 2 - dimensional subspace of real L_{4} is isometrically isomorphic to a subspace of real ℓ_{4}^{3}, hence N(2, 4, ℝ) = 3.

Recall that a sequence (x_{j})_{j=1}^{n} in an n - dimensional Banach space X is called an Auerbach basis provided there are linear functionals x_{1}^{∗}, …, x_{n}^{∗} in X^{∗} such that

∥x_{j}^{∗}∥ = ∥x_{j}∥ = x_{j}^{∗}(x_{j}) = 1 and x_{j}^{∗}(x_{k}) = 0 for j ≠ k (j, k = 1, …, n) .

It is well - known that every finite - dimensional Banach space admits an Auerbach basis (see e.g. DJT , Lemma 6.26).

Lemma 2.2. Let E be a 2 - dimensional subspace of real L_{4}(μ), μ any measure. Then there exist m_{E} ∈ 0, \frac{1}{3} and an Auerbach basis (f_{1}, f_{2}) of E such that

(2.7)  ∫ f_{1}^{2}f_{2}^{2} dμ = m_{E} .

Moreover, if (g_{1}, g_{2}) is any Auerbach basis of E such that m = ∫ g_{1}^{2}g_{2}^{2} dμ ∈ 0, \frac{1}{3} , then m = m_{E}.

Any Auerbach basis (f_{1}, f_{2}) satisfying (2.7) has the following two properties:

(2.8)  (f_{1}, f_{2}) is a symmetric basis for E .

(2.9) The ellipse of maximal area inscribed into B_{E} = {g ∈ E : ∥g∥_{L_{4}(μ)} ≤ 1} is given by

E_{E} = {g ∈ E : g = α_{1}f_{1} + α_{2}f_{2} and α_{1}^{2} + α_{2}^{2} ≤ 1} .

In addition, if 0 ≤ m_{E} < \frac{1}{3}, then the Auerbach basis satisfying (2.7) is unique. On the other hand, if m_{E} = \frac{1}{3}, then E is isometrically isomorphic to ℓ_{2}^{2}.

Proof. Using the Hahn - Banach Extension Theorem, the duality between L_{4}(μ) and L_{4/3}(μ) and the fact that for any 0 ≠ f ∈ L_{4}(μ) there is a unique g ∈ L_{4/3}(μ) with ∥g∥_{4/3} = 1 and ∫ gf dμ = ∥f∥_{4}, we infer that if (f_{1}, f_{2}) is an Auerbach basis for E, then the real - valued functions f_{1}^{3} and f_{2}^{3} represent the biorthogonal functionals to (f_{1}, f_{2}). Hence

(2.10)  ∫ f_{1}^{3}f_{2} dμ = ∫ f_{1}f_{2}^{3} dμ = 0 .

Next observe that if (f_{1}, f_{2}) is an Auerbach basis for E then, setting m := ∫ f_{1}^{2}f_{2}^{2}dμ and putting f̃_{1} := \frac{f_{1} + f_{2}}{\sqrt[4]{2 + 6m}} and f̃_{2} := \frac{f_{1} − f_{2}}{\sqrt[4]{2 + 6m}}, we get another Auerbach basis for E with

m̃ := ∫ f̃_{1}^{2}f̃_{2}^{2} dμ = \frac{1 − m}{1 + 3m} .

Clearly, min \{m, m̃\} ∈ 0, \frac{1}{3} because ∫ f_{1}^{2}f_{2}^{2}dμ ∈ 0, 1 by the Schwarz inequality. We have proved the existence of an Auerbach basis for E which satisfies (2.7) for some m_{E} ∈ 0, \frac{1}{3} .

Now pick α_{1}, α_{2} ∈ ℝ arbitrarily. It follows from (2.10) that

(2.11)  ∥α_{1}f_{1} + α_{2}f_{2}∥_{L_{4}(μ)}^{4} = α_{1}^{4} + α_{2}^{4} + 6α_{1}^{2}α_{2}^{2}m_{E} ,

and we get (2.8) since the right hand side is a symmetric function of the variables α_{1}^{2} and α_{2}^{2}. To prove (2.9), put

ψ(f, g) = α_{1}β_{1} + α_{2}β_{2} for f = α_{1}f_{1} + α_{2}f_{2} and g = β_{1}f_{1} + β_{2}f_{2}

and note that ψ is a non - degenerate and symmetric bilinear form on E×E satisfying ψ(g, g) ≥ 0 for all g ∈ E. The set E_{E} defined in (2.9) coincides with {g ∈ E: ψ(g, g) ≤ 1} and so it is an ellipse. Moreover, since 0 ≤ m_{E} ≤ \frac{1}{3}, we can use (2.11) to obtain

> ∥α_{1}f_{1} + α_{2}f_{2}∥_{L_{4}(μ)}^{4} = ψ(α_{1}f_{1} + α_{2}f_{2}, α_{1}f_{1} + α_{2}f_{2})^{2} − 6 · (\frac{1}{3} − m_{E})α_{1}^{2}α_{2}^{2}

(2.12)

≤ ψ(α_{1}f_{1} + α_{2}f_{2}, α_{1}f_{1} + α_{2}f_{2})^{2} .

Note that for equality in (2.12) it is necessary and sufficient that either m_{E} = \frac{1}{3}, or that m_{E} < \frac{1}{3} and α_{1}^{2}α_{2}^{2} = 0. It also follows ¿from (2.12) that E_{E} ⊂ B_{E}. Moreover, since (f_{1}, f_{2}) is an Auerbach basis, B_{E} is contained in the parallelogram

Q_{E} := {g ∈ E : g = α_{1}f_{1} + α_{2}f_{2} with max {|α_{1}|, |α_{2}|} ≤ 1} .

Then E_{E} is the ellipse of maximal area inscribed into Q_{E} since the affine transformation which takes f_{1} and f_{2} into the unit vectors (1, 0) and (0, 1) of ℝ^{2}, respectively, maps E_{E} onto the unit circle and Q_{E} onto the square −1, 1 × −1, 1 . But E_{E} ⊂ B_{E} ⊂ Q_{E} so that E_{E} is also the ellipse of maximal area inscribed into B_{E}.

Let now (f′_{1}, f′_{2}) be another Auerbach basis for E satisfying (2.7), with f′_{j} replacing f_{j} (j = 1, 2) and some m′_{E} ∈ 0, \frac{1}{3} replacing m_{E}. Using the f′_{j}’s, we define E′_{E} by (2.9) as before. Assume first that 0 ≤ m_{E} < \frac{1}{3}. Since the ellipse of maximal area inscribed into B_{E} is unique (see e.g. T-J , Corollary 5.11), we conclude that E′_{E} = E_{E}. By what was stated right after (2.12), it is immediate that the boundaries ∂E_{E}, ∂E′_{E} and ∂B_{E} satisfy

∂E_{E} ∩ ∂B_{E} = {±f_{1}, ±f_{2}} for m_{E} < 1/3 and ∂E′_{E} ∩ ∂B_{E} = {±f′_{1}, ±f′_{2}} for m′_{E} < 1/3 .

Hence the bases {f_{1}, f_{2}} and {f′_{1}, f′_{2}} conincide up to a permutation, and this implies m_{E} = m′_{E}.

If m_{E} = \frac{1}{3} or m′_{E} = \frac{1}{3}, then (2.12) is an identity for all α_{1}, α_{2} ∈ ℝ. We get B_{E} = E_{E} = E′_{E}, and so E is isometrically isomorphic to ℓ_{2}^{2}.

Keeping the above notation, we may now state:

Corollary 2.3. Suppose that E and E_{1} are 2 - dimensional subspaces of the real spaces L_{4}(μ) and L_{4}(μ_{1}), respectively. Then E and E_{1} are isometrically isomorphic if and only if m_{E} = m_{E_{1}}.

Proof. Apply the uniqueness of m_{E} and m_{E_{1}} and the formula (2.11).

Proof of Proposition 2.1. In view of Corollary 2.3, it suffices to establish

(2.13) For every m ∈ 0, \frac{1}{3} there exists a 2 - dimensional subspace E of real ℓ_{4}^{3} such that m_{E} = m.

For this we apply a continuity argument. Represent ℓ_{4}^{3} as ℝ^{3} with the corresponding canonical norm and let G_{2}^{3} be the set of unit balls B_{E} of 2 - dimensional subspaces E ⊂ ℓ_{4}^{3} considered as subsets of ℝ^{3}; so B_{E} = E ∩ B_{ℓ_{4}^{3}}. Let η( · , ·) be the Hausdorff distance between bounded subsets of ℝ^{3} relative to the standard Euclidean metric of the latter. Consider the metric spaces (G_{2}^{3}, η) and (E_{2}^{3}, η). The second is defined similar as the first with B_{E} replaced by E_{E}.

We claim that the function B_{E} ↦ m_{E} from (G_{2}^{3}, η) to ℝ is continuous.

Indeed, if lim_{n} η(B_{E_{n}}, B_{E})=0, then also lim_{n} η(E_{E_{n}}, E_{E})=0 for the associated ellipses. Now, if lim sup_{n} m_{E_{n}} < \frac{1}{3} then, by the uniqueness of the corresponding Auerbach bases satisfying (2.12), we also have lim_{n} η(∂B_{E_{n}} ∩ ∂E_{E_{n}}, ∂B_{E} ∩ ∂E_{E}) = 0, and lim_{n} m_{E_{n}} = m_{E} follows since, for each F ∈ G_{2}^{3}, m_{F} is expressed in terms of coordinates of the corresponding Auerbach basis of F. Suppose next that lim sup_{n} m_{E_{n}} = \frac{1}{3}. Passing to a subsequence if necessary we may assume that lim_{n} m_{E_{n}} = \frac{1}{3}. From (2.12) and a simple compactness argument applied in the space (E_{2}^{3}, η) it follows that the sequence (B_{E_{n}})_{n} tends to an ellipse. Thus m_{E} = \frac{1}{3}, and the claim is proved.

It is clear that if E_{0} is the subspace spanned by (1, 0, 0) and (0, 1, 0), then m_{E_{0}} = 0. Moreover, an easy computation reveals that if ϱ = 2^{−1/4}, σ = (18)^{−1/4} and E_{1} is the subspace spanned by (ϱ, ϱ, 0) and (σ, −σ, −2σ), then m_{E_{1}} = \frac{1}{3}. Since the ‘Grassmannian’ (G_{2}^{3}, η) is connected, (2.13) is an easy consequence of the above claim.

The proof of the counterpart of Proposition 2.1 in the complex case is even simpler. Let (Ω, Σ, μ) be an arbitrary measure space.

Proposition 2.4. Every 2 - dimensional subspace of complex L_{4}(μ) embeds isometrically into complex ℓ_{4}^{4}.

We begin with some notation and a few observations. Given f, g ∈ L_{4}(μ), we put

m(f, g) := ∫_{Ω}|f|^{2}|g|^{2}dμ  and  n(f, g) := | ∫_{Ω}f^{2}\overline{g}^{2}dμ| .

Clearly, if ∥f∥_{L_{4}(μ)} = ∥g∥_{L_{4}(μ)} = 1, then

0 ≤ n(f, g) ≤ m(f, g) ≤ 1 .

Next observe that if {f, g} is an Auerbach basis for a subspace of L_{4}(μ) then, for any α ∈ ℂ with |α| = 1, {αf, g} is also an Auerbach basis, m(f, g) = m(αf, g) and n(αf, g) = n(f, g). Choosing α so that α^{2} ∫_{Ω}f^{2}\overline{g}^{2}dμ = n(f, g), we get for arbitrary a, b ∈ ℂ,

(2.14)  ∥a(αf) + bg∥_{L_{4}(μ)}^{4} = |a|^{4} + |b|^{4} + 4|a|^{2}|b|^{2}m(f, g) + 2Re (a\overline{b})^{2}n(f, g) .

The arguments to prove this are similar to those which led to (2.11) because the complex - valued functions |f|^{2} \overline{f} and |g|^{2} \overline{g} are just the ones which represent the biorthogonal functionals to (f, g).

¿From (2.14) we obtain:

Corollary 2.5. Let μ_{1} and μ_{2} be arbitrary measures, and let {f_{j}, g_{j}} be an Auerbach basis for a subspace E_{j} of complex L_{4}(μ_{j}), j = 1, 2. If m(f_{1}, g_{1}) = m(f_{2}, g_{2}) and n(f_{1}, g_{1}) = n(f_{2}, g_{2}), then E_{1} and E_{2} are isometrically isomorphic.

We need one more lemma to prove Proposition 2.4.

Lemma 2.6. For every choice of 0 ≤ n ≤ m ≤ 1 there exists an Auerbach basis {f, g} for some 2 - dimensional subspace of complex ℓ_{4}^{4} such that m(f, g) = m and n(f, g) = n.

Proof. For 0 ≤ t, φ, ψ ≤ π/2 we put

f_{t,φ} = 2^{−1/4}(\sqrt{cos φ} , −\sqrt{cos φ} , e^{it}\sqrt{sin φ} , −e^{it}\sqrt{sin φ})
g_{t,ψ} = 2^{−1/4}(\sqrt{cos ψ} , \sqrt{cos ψ} , \sqrt{sin ψ} , \sqrt{sin ψ})

It is readily seen that {f_{t,φ}, g_{t,ψ}} is an Auerbach basis in its span in ℓ_{4}^{4}, and a simple computation reveals that

m(f_{t,φ}, g_{t,ψ}) = cos(φ − ψ)
n(f_{t,φ}, g_{t,ψ}) = | cos φ · cos ψ + e^{2it} · sin φ · sin ψ| .

Put

α := arccos m , φ := \frac{π}{4} + \frac{α}{2} and ψ := \frac{π}{4} − \frac{α}{2} .

Then 0 ≤ α ≤ π/2 , 0 ≤ (π/4) ± (α/2) ≤ π/2 , and m(f_{t,φ}, g_{t,ψ}) = cos α = m. Furthermore,

n(f_{t,φ}, g_{t,ψ}) = | cos (\frac{π}{4} + \frac{α}{2}) · cos (\frac{π}{4} − \frac{α}{2}) + e^{2it} · sin (\frac{π}{4} + \frac{α}{2}) · sin (\frac{π}{4} − \frac{α}{2})|

= |\frac{1 + e^{2it}}{2} · cos α| = cos^{2} t · cos α = m · cos^{2} t .

To get to the desired conclusion just take t := arccos \sqrt{\frac{n}{m}}.

The proof of Proposition 2.4 is now easy to complete.

Corollary 2.7. N(2, 4, ℂ) = 4 .

Proof. The upper estimate N(2, 4, ℂ) ≤ 4 follows from Proposition 2.4, while the lower estimate follows from the equality H(2, 4, ℂ) = 4 (see Kö , last paragraph on p. 54).

Remark. The counterpart of formula (2.8) (Lemma 2.2) is false in the complex case. From (2.14) we get:

An Auerbach basis {f, g} for a subspace of complex L_{4}(μ) is a (complex) unconditional basis of norm 1 iff n(f, g) = 0; in that case it is a (complex) symmetric basis of norm one.

3. The nonexistence of finite - dimensional Banach spaces isometrically universal for certain families of finite - dimensional Banach spaces

We begin with Klee’s criterion mentioned in the introduction.

For integers k, n with 1 ≤ k < n we put

c_{n,k}^{ℝ} := k(n − k)  and  c_{n,k}^{ℂ} := 2k(n − k) .

The topological dimension of a metrizable space M is denoted by tdim M (cf. HW , p. 24).

Klee’s Criterion 3.1. Let F be a family of k - dimensional Banach spaces. If there exists an n - dimensional Banach space which is isometrically universal for F then F, with the topology induced from BM_{k}, satisfies

tdim F ≤ c_{n,k}^{𝕂} .

In the Appendix we briefly describe Klee’s approach to this result.

The next theorem, stated in terms of Local Theory of Banach Spaces, is the main result of this section. Definitions and basic properties related to the notion of type and cotype of a Banach space can be found e.g. in DJT , LT and T-J .

Theorem 3.2. Let X be an infinite - dimensional Banach space. Assume that p(X) ∉ 2ℕ or q(X) ∉ 2ℕ, where

p(X) = sup {1 ≤ p ≤ 2 : X is of type p}
q(X) = inf {2 ≤ q ≤ ∞ : X is of cotype q} .

Then there is no finite - dimensional Banach space which is isometrically universal for BM_{2}(X).

The first step in the proof of Theorem 3.2 consists in a reduction to the case of the family

T_{p} := ⋃_{m≥3} BM_{2}(L_{p}^{m}) (p ∉ 2ℕ) .

Klee’s Criterion 3.1 will be applied to T_{p} when p ∉ 2ℕ, and this condition on p is in fact crucial. We need the following lemma which will lead us to

(3.1)  tdim BM_{2}(L_{p}^{m}) ≥ m − 2  ∀ m ∈ ℕ , m ≥ 3 .

For m ∈ ℕ, m ≥ 3, we define

Q_{m} := {a = (a_{j}) ∈ ℝ^{m} : a_{1} > 0 > a_{2} > … > a_{m}; ∑_{j} a_{j} = 0; ∥a∥_{L_{p}^{m}} = 1} ,

and we put, for a ∈ Q_{m},

E_{a} := span {1, a} .

We consider E_{a} as a subspace of L_{p}^{m}.

Lemma 3.3. Let 0 < p < ∞, p ∉ 2ℕ, and a, b ∈ Q_{m}. If there exists an isometric isomorphism u : E_{a} → E_{b}, then a = b and u = s_{0} · id_{E_{a}} for some s_{0} ∈ 𝕂 with |s_{0}| = 1.

Proof. Let s_{0}, t_{0}, s_{1}, t_{1} ∈ 𝕂 be such that u(1) = s_{0}1 + t_{0}b and u(a) = s_{1}1 + t_{1}b. Since p ∉ 2ℕ, u(1) has full support ( Har , Lemma 3.4), so that s_{0} + t_{0}b_{j} ≠ 0 for j = 1, …, m. We define a probability measure μ on the field of all subsets of {1, …, m} by

μ(A) := m^{−1} · ∑_{j∈A} |s_{0} + t_{0}b_{j}|^{p} (A ⊂ {1, …, m}) ,

and an isometric isomorphism w : L_{p}^{m} → L_{p}(μ) by

w(f) := ([ |s_{0} + t_{0}b_{j}| · sign (s_{0} + t_{0}b_{j})]^{−1} · f_{j})_{j=1}^{m} for f = (f_{j})_{1}^{m} ∈ L_{p}^{m} .

Accordingly, v = wu : E_{a} → L_{p}(μ) is an isometric embedding. Since obviously v(1) = 1, the Equimeasurability Lemma 0.1 yields μ_{a} = μ_{x} where

x = v(a) = ([ |s_{0} + t_{0}b_{j}| · sign (s_{0} + t_{0}b_{j})]^{−1}(s_{1} + t_{1}b_{j}))_{j=1}^{m} .

But μ_{x} = m^{−1} ∑_{j=1}^{m} |s_{0} + t_{0}b_{j}|^{p}δ_{x_{j}} and μ_{a} := m^{−1} ∑_{j=1}^{m} δ_{a_{j}}, hence 1 = |s_{0} + t_{0}b_{j}|^{p} for all j = 1, …, m. Since the b_{j}’s are real,

1 = |s_{0} + t_{0}b_{j}|^{2} = |s_{0}|^{2} + b_{j}^{2}|t_{0}|^{2} + 2b_{j}[ (Re s_{0})(Re t_{0}) + (Im s_{0})(Im t_{0})]

for all j = 1, …, m. Hence, if 1 ≤ j < k ≤ m, then

0 = (b_{j}^{2} − b_{k}^{2})|t_{0}|^{2} + 2(b_{j} − b_{k})[ (Re s_{0})(Re t_{0}) + (Im s_{0})(Im t_{0})]

and so b_{j} ≠ b_{k} implies

0 = |t_{0}|^{2}(b_{j} + b_{k}) + 2[ (Re s_{0})(Re t_{0}) + (Im s_{0})(Im t_{0})] .

Since ∑_{j=1}^{m} b_{j} = 0, summation over all pairs (j, k) with 1 ≤ k < j ≤ n yields

m(m − 1)[ (Re s_{0})(Re t_{0}) + (Im s_{0})(Im t_{0})] = 0 .

This in turn leads to 0 = |t_{0}|^{2}(b_{j} + b_{k}) for 1 ≤ j < k ≤ m. Taking j = 2 and k = 3, we get t_{0} = 0 from b_{2} < 0, b_{3} < 0, and so u(1) = s_{0}1. Since u is an isometry, |s_{0}| = 1 follows, and so x_{j} = (s_{1} + t_{1}b_{j})/s_{0} for each j = 1, …, m.

On the other hand, since μ_{a} = μ_{x}, there is a permutation π of {1, …, m} such that

a_{j} = x_{π(j)} = \frac{s_{1} + t_{1}b_{π(j)}}{s_{0}} for j = 1, …, m .

Using our hypothesis on a and b, we get by summation

0 = ∑_{j=1}^{m} a_{j} = \frac{s_{1}}{s_{0}} + t_{1} · ∑_{j=1}^{m} b_{π(j)} = \frac{s_{1}}{s_{0}} ,

whence s_{1} = 0, and so a_{j} = (t_{1}/s_{0})b_{π}(j) for all j = 1, …, m. But ∥a∥_{L_{p}^{m}} = ∥b∥_{L_{p}^{m}}, so that |t_{1}/s_{0}| = 1. Since m ≥ 3, some j_{0} ∈ {1, …, m} must be such that j_{0} > 1 and π(j_{0}) > 1, which forces a_{j_{0}} < 0 and b_{π(j_{0})} < 0. ¿From a_{j_{0}} = (t_{1}/s_{0})b_{π(j_{0})} we infer that t_{1}/s_{0} is positive, thus t_{1} = s_{0}. It follows that u(a) = s_{1} + t_{1}b = s_{0}b, and this allows us to conclude that u = s_{0} · id_{E_{a}}.

Corollary 3.4. If 1 ≤ p < ∞ is not an even integer, then

tdim BM_{2}(L_{p}^{m}) ≥ m − 2 ∀ m ∈ ℕ , m ≥ 3 .

Proof. For each a ∈ Q_{m}, let ϕ_{m}(a) ∈ BM_{2}(L_{p}^{m}) be the class generated by all subspaces of L_{p}^{m} which are isometrically isomorphic to E_{a}. We claim that the resulting map ϕ_{m} : Q_{m} → BM_{2}(L_{p}^{m}) is continuous.

To see why this is so, fix a, b ∈ Q_{m} and consider the isomorphism u_{a,b} : E_{a} → E_{b} defined by

u_{a,b}(s1 + ta) := s1 + tb .

Note that u_{a,b}^{−1} = u_{b,a}. Taking the inner product with respect to normalized counting measure (denoted by ( · | ·)), using the orthogonality of the functions 1 and a and applying Hölder’s inequality (with p^{∗} = \frac{p}{p−1}), we get

|t| · ∥a∥_{L_{2}^{m}}^{2} = |(s1 + ta|a)| ≤ ∥s1 + ta∥_{L_{p}^{m}} · ∥a∥_{L_{p^{∗}}^{m}} .

Let D = D(m) and d = d(m) be the norms of the formal identities L_{2}^{m} → L_{p^{∗}}^{m} and L_{2}^{m} → L_{p}^{m}, respectively. In particular, ∥a∥_{L_{p^{∗}}^{m}} ≤ D∥a∥_{L_{2}^{m}} and 1 = ∥a∥_{L_{p}^{m}} ≤ d∥a∥_{L_{2}^{m}}. Combining these three inequalities we easily obtain

|t| ≤ C · ∥s1 + ta∥_{L_{p}^{m}}

with C = D · d. Thus

∥s1 + ta − u_{a,b}(s1 + ta)∥_{L_{p}^{m}} = |t| · ∥a − b∥_{L_{p}^{m}} ≤ C · ∥a − b∥_{L_{p}^{m}} · ∥s1 + ta∥_{L_{p}^{m}}

and so ∥u_{a,b}(s1 + ta)∥_{L_{p}^{m}} ≤ (1 + C · ∥a − b∥_{L_{p}^{m}}) · ∥s1 + ta∥_{L_{p}^{m}}. Consequently,

∥u_{a,b} : E_{a} → E_{b}∥ ≤ 1 + C · ∥a − b∥_{L_{p}^{m}} .

Similarly, ∥u_{b,a} : E_{b} → E_{a}∥ ≤ 1 + C · ∥a − b∥_{L_{p}^{m}}, hence

d(E_{a}, E_{b}) ≤ (1 + C · ∥a − b∥_{L_{p}^{m}})^{2} .

This proves continuity of ϕ_{m}.

By Lemma 3.3, ϕ_{m} is one - to - one and so it induces a homeomorphism on every compact subset of Q_{m}.

Now observe that Q_{m} is a real Euclidean manifold depending on m parameters each taking values in an open interval, the parameters being constrained by the two conditions ∑_{j=1}^{m} |a_{j}|^{p} = m and ∑_{j=1}^{m} a_{j} = 0. It follows that every point in Q_{m} has a compact neighbourhood U such that tdim U = m − 2. Since ϕ_{m} induces a homeomorphism on U, we conclude that

m − 2 = tdim U = tdim ϕ_{m}(U) ≤ tdim BM_{2}(L_{p}^{m}) .

Remark. By the results in Be and Kl , the assertion of Corollary 3.4 is true for p = ∞ as well.

Proof of Theorem 3.2. By deep results of B. Maurey and G. Pisier MP and of J.L. Krivine Kr , every infinite - dimensional Banach space X contains, for every m ∈ ℕ and ε > 0, m - dimensional subspaces E_{ε}^{m} and F_{ε}^{m} such that d(E_{ε}^{m}, L_{p(X)}^{m}) < 1 + ε and d(F_{ε}^{m}, L_{q(X)}^{m}) < 1 + ε.

Assume now that there exists a finite - dimensional Banach space, say G, which is isometrically universal for all 2 - dimensional subspaces of X. Then, for every fixed integer m ≥ 3 and every ε > 0,

BM_{2}(E_{ε}^{m}) ⊂ BM_{2}(X) ⊂ BM_{2}(G) .

Since dim G < ∞, BM_{2}(G) is compact, and so we get

BM_{2}(L_{p(X)}^{m}) ⊂ BM_{2}(G)

by passing with ε to zero. Similarly,

BM_{2}(L_{q(X)}^{m}) ⊂ BM_{2}(G) .

An appeal to Klee’s Criterion 3.1 yields

max {tdim BM_{2}(L_{p(X)}^{m}), tdim BM_{2}(L_{q(X)}^{m})} ≤ tdim BM_{2}(G) ≤ c_{N,2}^{𝕂} ,

where N = dim G. If either p(X) or q(X) is not an even integer, then we get a contradiction with Corollary 3.4.

Corollary 3.5. Under the assumptions of Theorem 3.2, if k = 3, 4, …, then there is no finite - dimensional Banach space which is isometrically universal for all k - dimensional subspaces of X.

The proof of Theorem C is now straightforward.

Proof of Theorem C. Since the indices p(X) and q(X) are isomorphic invariants, we infer that if a Banach space X is isomorphic either to L_{p} or to ℓ_{p}, then p(X) = p(L_{p}) = p(ℓ_{p}) and q(X) = q(L_{p}) = q(ℓ_{p}). Now it follows from classical results of Orlicz (see e.g. LT , p. 73) that p(X) = p if 1 ≤ p ≤ 2, and q(X) = p if 2 ≤ p ≤ ∞. So the conclusion of Theorem C follows from Theorem 3.2 because p ∉ 2ℕ.

Remark. The concept of isometrically universal spaces generalizes in a natural fashion to quasi - Banach spaces. In such a setting, Theorem C can be extended to 0 < p < 1. For this we use the fact that if 0 < p < 1, then L_{p} contains a subspace isometric to ℓ_{1}, the latter taken in the metric induced by (∥ · ∥_{ℓ_{1}})^{p} (compare e.g. with DF , 24.5, Corollary 1).

The next proposition shows that the existence of a finite - dimensional Banach space which is isometrically universal for all k - dimensional subspaces of a given infinite - dimensional Banach space is rather exceptional from the viewpoint of isomorphic theory of Banach spaces. Proposition 3.6. Every infinite - dimensional Banach space is isomorphic to a Banach space Y such that there is no finite - dimensional Banach space which is isometrically universal for BM_{2}(Y).

Proof. Let Z be a subspace of X which admits a finite - dimensional decomposition (Z_{k})_{k∈ℕ} such that dim Z_{k} = k and d(Z_{k}, L_{2}^{k}) < 1 + 2^{−(k+1)} for each k ∈ ℕ. The existence of such a space Z goes back to the work of V.I. Gurariĭ Gu ; it follows from Dvoretzky’s Theorem on spherical sections (see e.g. DJT , 19.1). Pick scalars p_{k} so that 2 < p_{k} < 2 + 2^{−k} and d(L_{2}^{k}, L_{p_{k}}^{k}) < 1 + 2^{−k−1}, k ∈ ℕ. Then, for each k, there is an isomorphism u_{k} : Z_{k} → L_{p_{k}}^{k} such that

∥z∥ ≤ ∥u_{k}(z)∥_{L_{p_{k}}} ≤ (1 + 2^{−k+1})∥z∥ (z ∈ Z_{k}) .

Define a new norm on Z by

|||z||| := max {∥z∥, sup_{k} ∥u_{k}(z)∥} for z = ∑_{k=1}^{∞} z_{k} ∈ Z .

Since (Z_{k})_{k∈ℕ} is a finite - dimensional decomposition of Z, there is a constant C > 0 such that

(3.2)  ∥z∥ ≤ |||z||| ≤ C · ∥z∥ for all z ∈ Z .

By Pe , Proposition 1, the norm ||| · ||| extends to the norm on X, denoted by the same symbol, for which (3.2) is preserved for all z ∈ X. Put Y = (X, ||| · |||). Then Y is isomorphic to X, and the subspace (Z_{k}, ||| · |||) is isometrically isomorphic to L_{p_{k}}^{k}. Thus tdim BM_{2}((Z_{k}, ||| · |||)) ≥ k − 2 by Corollary 3.4, and an inessential modification of the proof of Theorem 3.2 yields the desired conclusion.

Being more careful one can show that, given ε > 0, the constant C in (3.2) can be replaced by 1 + ε.

Appendix: A Guide to a Proof of Klee’s Criterion 3.1

We begin by introducing some notation. Let us denote by

• C^{n} the family of all compact absolutely convex bodies in 𝕂^{n},

• G_{k}^{n} the Grassmannian of k - dimensional linear subspaces of 𝕂^{n} (1 ≤ k ≤ n),

• C_{k}^{n} the collection of all C ⊂ 𝕂^{n} such that C = B ∩ H for some B ∈ C^{n} and some H ∈ G_{k}^{n},

• η the Hausdorff distance between compact subsets of 𝕂^{n} induced by the standard Euclidean metric in 𝕂^{n},

• γ the metric on G_{k}^{n} defined by γ(H_{1}, H_{2}) := η(H_{1} ∩ U, H_{p} ∩ U) where U = B_{ℓ_{2}^{n}} is the standard Euclidean unit ball in 𝕂^{n},

• GL_{n} the group of nonsingular linear transformations of 𝕂^{n},

•  C̃^{n} the family of equivalence classes of affinely equivalent members of C^{n}.

Here B, C ∈ C^{n} are said to be affinely equivalent if C = σ(B) for some σ ∈ GL_{n}.

We consider two metrics on C̃_{n}. The first one is again the Banach - Mazur metric δ( · , ·); recall that it is derived in the canonical fashion from the pseudometric on C^{n} given by

δ(B_{1}, B_{2}) := log d(B_{1}, B_{2}) where d(B_{1}, B_{2}) := inf t

for B_{1}, B_{2} ∈C^{n}; the infimum is taken over all t≥1 and σ ∈GL_{n} such that B_{1} ⊂σ(B_{2})⊂t·B_{1}.

The second is the Macbeath metric β( · , ·) which is derived analogously from the pseudometric on C^{n} given by

β(B_{1}, B_{2}) := log b(B_{1}, B_{2}) where b(B_{1}, B_{2}) := inf \frac{vol (σ_{1}(B_{1})}{vol (B_{2})} · \frac{vol (σ_{2}(B_{2}))}{vol (B_{1})} ,

for B_{1}, B_{2} ∈ C^{n}; here vol ( ·) denotes the n - dimensional volume function, and the infimum extends over all σ_{1}, σ_{2} ∈ GL_{n} such that B_{2} ⊂ σ_{1}(B_{1}) and B_{1} ⊂ σ_{2}(B_{2}).

We rely on a number of facts. (I) The metric space (C̃^{n}, δ) is isometric to BM_{n} under the metric used before; the isometry

assigns to each equivalence class of B ∈ C^{n} the equivalence class of Banach spaces isometrically isomorphic to (𝕂^{n}, B) := (𝕂^{n}, ∥ · ∥_{B}); here ∥ · ∥_{B} is the gauge functional of B.

(II) The metric space (C̃^{n}, β) is compact; cf. Mc . (III) For B_{1}, B_{2} ∈ C^{n}, we have β(B_{1}, B_{2}) ≤ 2nδ(B_{1}, B_{2}). Therefore the spaces BM_{n}, alias

(C̃^{n}, δ), and (C̃^{n}, β) are canonically homeomorphic.

Each B ∈ C^{n} gives rise to the n - dimensional Banach space E = (𝕂^{n}, B) which we use to define the map f_{E} : G_{k}^{n} → C_{k}^{n} by

f_{E}(H) := H ∩ B for H ∈ G_{k}^{n} .

Let q : C_{k}^{n} → C̃^{k} be the map which associates with each B ∈ C_{k}^{n} its affine equivalence class. The crucial observation is that the composition (IV) f̃_{E} := q ∘ f_{E} : (G_{k}^{n}, γ) ⟶ (C̃^{k}, β) is Lipschitzian. This is a consequence of two results: (IVa) f_{E} : (G_{k}^{n}, γ) ⟶ (C_{k}^{n}, η) is Lipschitzian and (IVb) q : (C_{k}^{n}, η) ⟶ (C̃^{k}, β) is locally Lipschitzian. As for (IVa), use Kl , Lemma 2.2, combined with Kl , Proposition 2.1, whereas (IVb) can be obtained by using Kl , Theorem 2.7 in a special case.

Write now hdim (M, ϱ) for the Hausdorff dimension (see HW , p. 107) of a metric space (M, ϱ). Directly from the definition of that dimension we get: (V) If (M_{1}, ϱ_{1}) is another metric space and f : (M, ϱ) → (M_{1}, ϱ_{1}) is Lipschitzian, then

hdim (M_{1}, ϱ_{1}) ≤ hdim (M, ϱ) .

On the other hand we have, for every separable metric space (M, ϱ),

(VI) tdim (M, ϱ) ≤ hdim (M, ϱ)

(see HW , Ch.VII, and Sz ). Finally (cf. Kl , Corollary 3.2 for the real case),

(VII) hdim (G_{k}^{n}, γ) = c_{n,k}^{𝕂} .

Klee’s result can now be proved by assembling these informations in an appropriate way.

Proof of Klee’s Criterion 3.1. Let F ⊂BM_{k} be a family of Banach spaces and suppose that E is an n - dimensional Banach space which is isometrically universal for F. Then F ⊂f̃_{E}(G_{k}^{n}), and so

tdim F ≤ tdim (f̃_{E}(G_{k}^{n}), δ) = tdim (f̃_{E}(G_{k}^{n}), β)  by (III)
≤ hdim (f̃_{E}(G_{k}^{n}), β)  by (VI)
≤ hdim (G_{k}^{n}, γ)  by (IV) and (V)
= c_{n,k}^{𝕂} .  by (VII)

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F. Delbaen  H. Jarchow  A. Pełczyński
Department Mathematik  Institut für Mathematik  Institute of Mathematics
ETH Zürich  Universität Zürich  Polish Academy of Sciences
Rämistr. 101  Winterthurerstr. 190  8, ul.Sniadeckich
CH 8092 Zürich  CH 8057 Zürich  PL00-950 Warszawa
Switzerland  Switzerland  Poland
delbaen@math.ethz.ch  jarchow@math.unizh.ch  olek@impan.gov.pl

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