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

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

For an arbitrary topological group G any compact G-dynamical system (G, X) can be linearly G-represented as a weak∗-compact subset of a dual Banach space V ∗. As was shown in [45] the Banach space V can be chosen to be reflexive iff the metric system (G, X) is weakly almost periodic (WAP). In this paper we study the wider class of compact G-systems which can be linearly represented as a weak∗-compact subset of a dual Banach space with the Radon-Nikodým property. We call such a system a Radon-Nikodým system (RN). One of our main results is to show that for metrizable compact G-systems the three classes: RN, HNS (hereditarily not sensitive) and HAE (hereditarily almost equicontinuous) coincide. We investigate these classes and their relation to previously studied classes of G-systems such as WAP and LE (locally equicontinuous). We show that the Glasner-Weiss examples of recurrent-transitive locally equicontinuous but not weakly almost periodic cascades are actually RN. Using fragmentability and Namioka’s theorem we give an enveloping semigroup characterization of HNS systems and show that the enveloping semigroup E(X) of a compact metrizable HNS G-system is a separable Rosenthal compact, hence of cardinality ≤ 2א0 . We investigate a dynamical version of the Bourgain-Fremlin-Talagrand dichotomy and a dynamical version of Todorcevic dichotomy concerning Rosenthal compacts.

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Hereditarily non-sensitive dynamical systems and linear representations

Enveloping semigroups in topological dynamics

Essentially Smooth Lipschitz Functions

Canonical Examples of Weak Asplund Spaces. Properties of Gateaux Differentiability Spaces and Weak Asplund Spaces. Stegall's Classes. Two More Concrete Classes of Banach Spaces that Lie in . Fragmentability. "Long Sequences" of Linear Projections. Vasak Spaces and Gul'ko Compacta. A Characterization of WCG Spaces and of Eberlein Compacta. Main Open Questions and Problems. References. Index.

Marián J. Fabian

Papers

Gâteaux differentiability of convex functions and topology : weak asplund spaces