Haskell P. Rosenthal

On the subspaces ofL p (p>2) spanned by sequences of independent random variables

Let 2<p<∞. The Banach space spanned by a sequence of independent random variables inLp, each of mean zero, is shown to be isomorphic tol2,lp,l2⊕lp, or a new spaceXp, and the linear topological properties ofXp are investigated. It is proved thatXp is isomorphic to a complemented subspace ofLp and another uncomplemented subspace ofLp, whence there exists an uncomplemented subspace oflp isomorphic tolp. It is also proved thatXp is not isomorphic to the previously knownℒp spaces.

On bases, finite dimensional decompositions and weaker structures in Banach spaces

This is an investigation of the connections between bases and weaker structures in Banach spaces and their duals. It is proved, e.g., thatX has a basis ifX* does, and that ifX has a basis, thenX* has a basis provided thatX* is separable and satisfies Grothendieck’s approximation property; analogous results are obtained concerning π-structures and finite dimensional Schauder decompositions. The basic results are then applied to show that every separableℒp space has a basis.

Point-Wise Compact Subsets of the First Baire Class

Let X be a complete separable metric space. Various characterizations of point-wise compact subsets of the first Baire class of real-valued functions on X are obtained. For example, it is proved that a compact subset is sequentially compact in the topology of point-wise convergence, and moreover (in the case where it is additionally uniformly bounded) that it is compact with respect to the topology induced by the set of Borel probability measures on X. The results are applied to show that 11 imbeds in a separable Banach space B provided there exists a bounded sequence in B** which has no weak*-convergent subsequence.

Descriptive Set Theory and Banach Spaces

Abstract The goal of this chapter is to reveal the power of descriptive set theory in penetrating the structure of Banach spaces. The chapter is divided into three subchapters, each with its own introduction. Subchapters one, two and three were mostly written by the second, third, and first authors, respectively. Space limitations forced us to leave out many fundamental results and on-going research in this exciting interface. We do hope, however, that our article gives a strong flavor of this aspect of Banach space theory.

Chapter 36 – The Banach Spaces C(K)

Table of Contents 1. Introduction. 2. The isomorphic classification of separable C(K) spaces. A. Milutins Theorem. B. C(K) spaces with separable dual via the Szlenk index 3. Some Banach space properties of separable C(K) spaces. A. Weak injectivity. B. c 0-saturation of spaces with separable dual. C. Uncomplemented embeddings of C([0, 1]) and C(ω ω +) in themselves. 4. Operators on C(K)-spaces.

Applications of the theory of semi-embeddings to Banach space theory

Let X and Y be Banach spaces and T:X → Y an injective bounded linear operator. T is called a semi-embedding if T maps the closed unit ball of X to a closed subset of Y. (This concept was introduced by Lotz, Peck, and Porta, Proc. Edinburgh Math. Soc. 22 (1979), 233–240.) It is proved that if X semi-embeds in Y, and X is separable, then X has the Radon-Nikodym property provided Y does. It is shown that if L1 semi-embeds in Y, then Y fails the Schur property and contains a subspace isomorphic to l1. As a consequence of the proof, it is shown that if X is a subspace of L1, either L1 embeds in X or l1 embeds in L1X. The simpler result that L1 does not semi-embed in c0 is treated separately. This result is used to deduce the classic result of Menchoff that there exists a singular probability measure on the circle with Fourier coefficients vanishing at infinity. Some generalizations of the notion of semi-embedding are given, and several complements and open questions are discussed.

A characterization of Banach spaces containing

It is proved that a Banach space contains a subspace isomorphic to 11 if (and only if) it has a bounded sequence with no weak-Cauchy subsequence. The proof yields that a sequence of subsets of a given set has a subsequence that is either convergent or Boolean independent. A bounded sequence of elements (fn) in a Banach space B is said to be equivalent to the usual 11-basis provided there is a a > 0 so that for all n and choices of scalars c, ... ,Cn)

On factors ofC([0, 1]) with non-separable dual

LetC denote the Banach space of scalar-valued continuous functions defined on the closed unit interval. It is proved that ifX is a Banach space andT:C→X is a bounded linear operator withT*X* non-separable, then there is a subspaceY ofC, isometric toC, such thatT|Y is an isomorphism. An immediate consequence of this and a result of A. Pelczynski, is that every complemented subspace ofC with non-separable dual is isomorphic (linearly homeomorphic) toC.

On the structure of non-dentable closed bounded convex sets

A self-contained proof is given of the following result. Theorem. Let K be a non-dentable closed bounded convex nonempty subset of a Banach space X so that K equals the closed convex hull of its weak-to-norm points of continuity. There exists a nonempty subset W of K satisfying W is non-dentable closed convex and the weak and norm topologies on W coincide. (∗) Moreover there exists a closed linear subspace Y of X so that Y has a Finite-Dimensional Decomposition and a nonempty convex subset W satisfying (∗). (Any bounded set W satisfying (∗) has no extreme points.) This yields the recent discovery of W. Schachermayer that a closed bounded convex subset of a Banach space has the Radon-Nikodým Property (RNP) provided it has the Convex Point of Continity Property (CPCP) and the Krein-Milman Property (KMP), and the CPCP case of the earlier discovery of J. Bourgain that a Banach space has the RNP provided every subspace with a Finite-Dimensional Decomposition has the RNP. The proof is a variation and crystallization of the original arguments for these discoveries. In particular, the proof uses the concept of convex sets with small combinations of slices. This is a refinement of the concept of strong regularity, introduced by N. Ghoussoub, G. Godefroy, and B. Maurey; both concepts crystallize ideas appearing in Bourgains work. The proof also yields further results concerning the structure of non-dentable convex sets.

Weak∗-Polish Banach spaces

Abstract A Banach space X which is a subspace of the dual of a Banach space Y is said to be weak ∗ -Polish proviced X is separable and the closed unit ball of X is Polish in the weak ∗ -topology. X is said to be Polish provided it is weak ∗ -Polish when regarded as a subspace of X ∗∗ . A variety of structural results are obtained for Banach spaces with the Point of Continuity Property (PCP), using the unifying concept of weak ∗ -Polish Banach spaces. It is proved that every weak ∗ -Polish Banach space contains an (infinite-dimensional) weak ∗ -closed subspace. This yields the result of Edgar-Wheeler that every Polish Banach space has a reflexive subspace. It also yields the result of Ghoussoub-Maurey that every Banach space with the PCP has a subspace isomorphic to a separable dual, in virtue of the following known result linking the PCP and weak ∗ -Polish Banach spaces: A separable Banach space has the PCP if (Edgar-Wheeler) and only if (Ghoussoub-Maurey) it is isometric to a weak ∗ -Polish subspace of the dual of some separable Banach space. (For completeness, a proof of this result is also given.) It is proved that if X and Y are Banach spaces with X ⊂ Y ∗ , then X is weak ∗ -Polish if and only if X has a weak ∗ -continuous boundedly complete skipped-blocking decomposition. Combined with the above-mentioned linkage, this characterization of weak ∗ -Polish Banach spaces gives the known result that a separable Banach space has the PCP if (Bourgain-Rosenthal) and only if (Ghoussoub-Maurey) it has the boundedly complete skipped blocking property. (The proof of the characterization is a variation of the original arguments for this result.) The characterization also yields that a Banach space is Polish if and only if it has the reflexive skipped-blocking property. It is moreover demonstrated that a Banach space has the PCP provided it has the PC-skipped-blocking property. In particular, this yields an exposition of the result of Bourgain that a space has the PCP provided each of its subspaces with an FDD has the PCP. It is proved that the PCP is a three-space property, thus completing the solution to a problem posed by Edgar-Wheeler; that is, if Y and X are Banach spaces with Y ⊂ X , then X has the PCP provided both Y and X Y have the PCP. It is also shown that a Banach space X has a separable dual provided it has the dual-separable skipped-blocking property. The proof yields the dividend that X ∗ is separable provided [x j ] ∗ is separable for every basic sequence ( x j ) in X .