Preface * 1 Basic Concepts in Banach Spaces * 2 Hahn-Banach and Banach Open Mapping Theorems * 3 Weak Topologies * 4 Locally Convex Spaces * 5 Structure of Banach Spaces * 6 Schauder Bases * 7 Compact Operators on Banach Spaces * 8 Differentiability of Norms * 9 Uniform Convexity * 10 Smoothness and Structure * 11 Weakly Compactly Generated Spaces * 12 Topics in Weak Toplogy * References * Index

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Functional Analysis and Infinite-Dimensional Geometry

We construct a Banach space that does not contain any infinite un- conditional basic sequence and investigate further properties of this space. For example, it has no subspace that can be written as a topological direct sum of two infinite-dimensional spaces. This property implies that every operator on the space is a strictly singular perturbation of a multiple of the identity. In par- ticular, it is either strictly singular or Fredholm with index zero. This implies that the space is not isomorphic to any proper subspace. TRINITY COLLEGE, CAMBRIDGE, CB2 1TQ ENGLAND Current address: Department of Mathematics, University College London, Gower Street, Lon- don WC1E 6BT, England E-mail address: wtg l O@phx.cam.ac.uk UNIVERSIT#, PARIS VII, U.F.R. DE MATHtMATIQUES, 2 PLACE JUSSIEU, 75251 PARIS CEDEX 05, FRANCE Current address: Equipe d'Analyse et Mathematiques Appliquees, Universite de Mamne la Vallee, 93160 Noisy le Grand, France E-mail address: maurey@logiquejussieu.fr This content downloaded from 157.55.39.105 on Fri, 07 Oct 2016 04:30:48 UTC All use subject to http://about.jstor.org/terms

/pdf/the-unconditional-basic-sequence-problem-4q8bizerue.pdf

The unconditional basic sequence problem

(A) In this paper we study some connections between the Fraisse theory of amalgamation classes and ultrahomogeneous structures, Ramsey theory, and topological dynamics of automorphism groups of countable structures. A prime concern of topological dynamics is the study of continuous actions of (Hausdorff) topological groups G on (Hausdorff) compact spaces X.

/pdf/fraisse-limits-ramsey-theory-and-topological-dynamics-of-bdb8ujgtuy.pdf

Fraïssé Limits, Ramsey Theory, and topological dynamics of automorphism groups

We construct a Banach space that does not contain any infinite unconditional basic sequence.

Like differentiability, convexity is a natural and powerful property of functions that plays a significant role in many areas of mathematics, both pure and applied. It ties together notions from topology, algebra, geometry and analysis, and is an important tool in optimization, mathematical programming and game theory. This book, which is the product of a collaboration of over 15 years, is unique in that it focuses on convex functions themselves, rather than on convex analysis. The authors explore the various classes and their characteristics and applications, treating convex functions in both Euclidean and Banach spaces. The book can either be read sequentially for a graduate course, or dipped into by researchers and practitioners. Each chapter contains a variety of specific examples, and over 600 exercises are included, ranging in difficulty from early graduate to research level.

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Convex Functions: Constructions, Characterizations and Counterexamples

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The distortion problem

Let B be a separable Banach space and let X=B
* be separable. We prove that if B has finite Szlenk index (for all e > 0) then B can be renormed to have the weak* uniform Kadec-Klee property. Thus if e > 0 there exists δ (e) > 0 so that if x
n is a sequence in the ball of X converging ω* to x so that 
$$\lim \inf _{n \to \infty } \left\| {x_n - x} \right\| \geqslant \varepsilon {\text{ then }}\left\| x \right\| \leqslant 1 - \delta (\varepsilon )$$

 . In addition we show that the norm can be chosen so that δ (e) ≥ cep for some p 0.

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On Asymptotic Structure, the Szlenk Index and UKK Properties in Banach Spaces

An infinite dimensional notion of asymptotic structure is considered. This notion is developed in terms of trees and branches on Banach spaces. Every countably infinite countably branching tree T of a certain type on a space X is presumed to have a branch with some property. It is shown that then X can be embedded into a space with an FDD (E i ) so that all normalized sequences in X which are almost a skipped blocking of (E i ) have that property. As an application of our work we prove that if X is a separable reflexive Banach space and for some 1 0, there exists a subspace of X having finite codimension which C 2 + e embeds into the l p sum of finite dimensional spaces.

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Trees and branches in Banach spaces

We introduce two new local l1-indices of the same type as the Bourgain l1-index; the l+1-index and the l+1-weakly null index. We show that the l+1-weakly null index of a Banach space X is the same as the Szlenk index of X, provided X does not contain l1. The l+1-weakly null index has the same form as the Bourgain l1-index: if it is countable it must take values ωα for some α<ω1. The different l1-indices are closely related and so knowing the Szlenk index of a Banach space helps us calculate its l1-index, via the l+1-weakly null index. We show that I(C(ωωα))=ω^1+α+1.

The Szlenk index and local 1-indices

Certain subclasses of $B_1(K)$, the Baire-1 functions on a compact metric space $K$, are defined and characterized. Some applications to Banach spaces are given.

Edward Odell

Papers

The distortion problem

On Asymptotic Structure, the Szlenk Index and UKK Properties in Banach Spaces

Trees and branches in Banach spaces

The Szlenk index and local 1-indices

On certain classes of Baire-1 functions with applications to Banach space theory