Petr Habala

Finite Representability of ℓp in Quotients of Banach Spaces

A typical result of the paper states that if X is a Banach space with a basis and for some 1≤p≤q≤∞, the spaces ℓp and ℓq are finitely block representable in every block subspace of X, then every block subspace of X admits a block quotient Z such that for every rℓ[p,q], the space ℓr is finitely block representable in Z. Results of a similar nature are also established for ℓNp-block-sequences and asymptotic spaces.

Basic Concepts in Banach Spaces

In this chapter we introduce basic notions and concepts in Banach space theory. As a rule we will work with real scalars, only in a few instances, e.g., in spectral theory, we will use complex scalars.

Dentability and Differentiability

The main topic of the present chapter is the dentability of bounded sets and the closely related Radon–Nikodým property (RNP) of Banach spaces. This property has several equivalent characterizations and applications. In particular, Asplund spaces are characterized by the Radon–Nikodým property of their dual spaces. As another application, we show that Lipschitz mappings from separable Banach spaces into Banach spaces with RNP are at some points Gâteaux differentiable.

Finite-Dimensional Spaces

The interplay between the structure of an infinite-dimensional Banach space and properties of its finite-dimensional subspaces belongs to the subject of the local theory of Banach spaces. It is a vast and deep part of Banach space theory intimately related to probability and combinatorics. Our goal is to familiarize the reader with some of its basic notions and results that are accessible without the use of deep probabilistic tools.

Structure of Banach Spaces

The focus of the study in the present chapter is on Banach spaces containing subspaces isomorphic to c 0 or l 1. We prove Sobczyk’s theorem on complementability of c 0 in separable overspaces, lifting property of l 1 and Pelczynski’s characterization of separable Banach spaces containing l 1. We present Rosenthal’s l 1 theorem, Odell–Rosenthal theorem and the Rosenthal–Bourgain–Fremlin–Talagrand theory of Baire-1 functions on Polish spaces.

Hahn—Banach and Banach Open Mapping Theorems

The Hahn-Banach theorem, in the geometrical form, states that a closed and convex set can be separated from any external point by means of a hyperplane. This intuitively appealing principle underlines the role of convexity in the theory. It is the first, and most important, of the fundamental principles of functional analysis. The rich duality theory of Banach spaces is one of its direct consequences. The second fundamental principle, the Banach open mapping theorem, is studied in the rest of the chapter.

Higher Order Smoothness

In this chapter, we will first discuss the properties of smoothness in l p spaces and in Hilbert spaces. Then we study spaces that have countable James boundary in connection with their higher order smoothness, and its applications. In particular, we study spaces of continuous functions on countable compact spaces.

Basics in Nonlinear Geometric Analysis

In this chapter, we begin by proving the Brouwer and the Schauder fixed-point theorems. Then we turn to results on homeomorphisms of convex sets and spaces. We prove Keller’s theorem on homeomorphism of infinite-dimensional compact convex sets in Banach spaces to \({\mathbb I}^{{\mathbb N}}\). We also prove the Kadec theorem on the homeomorphism of every separable reflexive space to a Hilbert space. Then we prove some results on uniform, in particular Lipschitz, homeomorphisms.

Weakly Compactly Generated Spaces

In this chapter we study weakly compact operators and the related class of Banach spaces that are generated by weakly compact sets (i.e., weakly compactly generated spaces, in short WCG spaces). We focus on their decomposition properties, renormings, and on the topological properties of their dual spaces. We prove that WCG spaces are generated by reflexive spaces. Then we study absolutely summing operators and the Dunford–Pettis property.

Topics in Weak Topologies on Banach Spaces

In this chapter we study the weak and weak* topologies of Banach spaces in more detail. We discuss several types of compacta (Eberlein, uniform Eberlein, scattered, Corson, and more), weakly Lindelof determined spaces and properties of tightness in weak topologies. We discuss some applications in the structural properties of some Banach spaces.