Topics in Banach space theory.

*(English)*Zbl 1094.46002
Graduate Texts in Mathematics 233. Berlin: Springer (ISBN 0-387-28141-X/hbk). xi, 373 p. (2006).

This book gives a self-contained overview of the fundamental ideas and basic techniques in modern Banach space theory. The authors’ approach is based on the analysis of Schauder bases and basic sequences in Banach spaces, first of all, in classical Banach spaces \(\ell_p \;(1 \leq p \leq \infty)\), \(c_0\), \(L_p \;(1 \leq p \leq \infty)\), and \(C(K)\). This approach allows them to present many fine results (of course, not all) about the geometry of Banach spaces, related primarily to the isomorphic structure of Banach spaces. In this book one can find a systematic and coherent account of numerous theorems and examples obtained by many remarkable mathematicians.

The book consists of 13 short chapters. Chapter 1, Bases and Basic Sequences, gives basic definitions and general results about Schauder bases and basic sequences (bases in subspaces); the chapter is completed with a short and transparent proof of the Eberlein-Šmulian theorem. Chapter 2, The Classical Sequences Spaces, presents the theory of spaces \(\ell_p\) \((1 \leq p \leq \infty)\) and \(c_0\); the main focus is on complemented subspaces in these spaces. The next Chapter 3, Special Types of Bases, deals with unconditional, boundedly-complete, and shrinking bases; here the reader can also find the description of the James space \({\mathcal J}\). Chapter 4, Banach spaces of continuous functions, considers basic properties of this space, in particular, the Goodner-Nachbin theorem and spaces of continuous functions on countable and uncountable compact metric spaces. Chapter 5, \(L_1(\mu)\)-Spaces and \(C(K)\)-Spaces, is devoted to weakly compact sets and weakly compact operators and the Dunford-Pettis property. Chapter 6, The \(L_p\)-spaces for \(1 \leq p < \infty\), deals with the Haar basis in \(L_p\), estimating expressions of type \(\| \varepsilon_1x_1 + \ldots + \varepsilon_nx_n\| \) \( (\varepsilon_i \in \{-1,1\})\), and subspaces of \(L_p\).

Chapter 7, Factorization Theory, is devoted to Maurey-Nikishin factorization theorems, factorization through Hilbert spaces, and the Kwapień-Maurey results for type-2-spaces. In Chapter 8, Absolutely Summing Operators, the classical Grothendieck inequality and absolutely summing operators are studied. Chapter 9, Perfectly Homogeneous Bases and Their Applications, is devoted to a characterization of the canonical bases of \(\ell_p \;(1 \leq p \leq \infty)\) and \(c_0\) due to Zippin and the Lindenstrauss-Pelczyński theorem on the uniqueness of the unconditional bases in \(c_0\), \(\ell_1\), and \(\ell_2\); also the Pelczyński-Singer theorem on the existence of conditional bases in any Banach space with a basis is proven. Chapter 10, \(\ell_p\)-Subspaces of Banach Spaces, presents the classical Rosenthal theorem on necessary and sufficient conditions for \(\ell_1\) to be isomorphic to a subspace of a given Banach space; here one can find a desription of Tsirelson’s space.

Chapter 11, Finite Representability of \(\ell_p\)-Spaces, is devoted to geometry of Banach spaces; here one can find the principle of local reflexivity and Krivine’s and Dvoretzky’s (qualitative variant) theorems. The next chapter 12, An Introduction to Local Theory, deals with a quantative aspect of the Dvoretzky theorem (in particular, important Milman’s estimates), the Kadets-Snobar theorem about the \(\sqrt{n}\)-complementarity of \(n\)-dimensional subspaces of Banach spaces, and, finally, the Lindenstrauss-Tzafriri theorem on the characterization of a Hilbert space in terms of complemented subspaces. The last chapter 13, Important Examples of Banach Spaces, is devoted to some cumbersome examples of Banach spaces playing a fundamental role in understanding this excellent branch of Functional Analysis. The first 12 chapters contain special problem sections, presenting altogether 122 problems. The main part of them deals with additional material, related to the basic substance of the book, however, left out of the main body of the text. Among them are: the bounded approximation property, Orlicz sequence spaces, the Stone-Weierstrass theorem, the Amir-Cambern theorem, the James distortion theorem, superreflexivity, and many others.

At the end of the book, seven “supplements” are given. They contain fundamental definitions and results of linear functional analysis, headlined: Fundamental Notions, Elementary Hilbert Space Theory, Main Features of Finite-Dimensional Spaces, Cornerstone Theorems of Functional Analysis, Convex Sets and Extreme Points, The Weak Topology, Weak Compactness of Sets and Operators. The book also contains a List of Symbols, References, and an Index.

The book grew out of a one-semester course (in 2001) and subsequent two-semester courses (in 2004-2005) given by Nigel J. Kalton at the University of Missouri–Columbia. It is intended for graduate students and specialists in classical functional analysis. To my mind, this book is a good complement to other books devoted to geometry of Banach spaces, and I think that every mathematician who is interested in geometry of Banach spaces should, at least, look over this book. Undoubtedly, the book will be a useful addition to any mathematical library.

The book consists of 13 short chapters. Chapter 1, Bases and Basic Sequences, gives basic definitions and general results about Schauder bases and basic sequences (bases in subspaces); the chapter is completed with a short and transparent proof of the Eberlein-Šmulian theorem. Chapter 2, The Classical Sequences Spaces, presents the theory of spaces \(\ell_p\) \((1 \leq p \leq \infty)\) and \(c_0\); the main focus is on complemented subspaces in these spaces. The next Chapter 3, Special Types of Bases, deals with unconditional, boundedly-complete, and shrinking bases; here the reader can also find the description of the James space \({\mathcal J}\). Chapter 4, Banach spaces of continuous functions, considers basic properties of this space, in particular, the Goodner-Nachbin theorem and spaces of continuous functions on countable and uncountable compact metric spaces. Chapter 5, \(L_1(\mu)\)-Spaces and \(C(K)\)-Spaces, is devoted to weakly compact sets and weakly compact operators and the Dunford-Pettis property. Chapter 6, The \(L_p\)-spaces for \(1 \leq p < \infty\), deals with the Haar basis in \(L_p\), estimating expressions of type \(\| \varepsilon_1x_1 + \ldots + \varepsilon_nx_n\| \) \( (\varepsilon_i \in \{-1,1\})\), and subspaces of \(L_p\).

Chapter 7, Factorization Theory, is devoted to Maurey-Nikishin factorization theorems, factorization through Hilbert spaces, and the Kwapień-Maurey results for type-2-spaces. In Chapter 8, Absolutely Summing Operators, the classical Grothendieck inequality and absolutely summing operators are studied. Chapter 9, Perfectly Homogeneous Bases and Their Applications, is devoted to a characterization of the canonical bases of \(\ell_p \;(1 \leq p \leq \infty)\) and \(c_0\) due to Zippin and the Lindenstrauss-Pelczyński theorem on the uniqueness of the unconditional bases in \(c_0\), \(\ell_1\), and \(\ell_2\); also the Pelczyński-Singer theorem on the existence of conditional bases in any Banach space with a basis is proven. Chapter 10, \(\ell_p\)-Subspaces of Banach Spaces, presents the classical Rosenthal theorem on necessary and sufficient conditions for \(\ell_1\) to be isomorphic to a subspace of a given Banach space; here one can find a desription of Tsirelson’s space.

Chapter 11, Finite Representability of \(\ell_p\)-Spaces, is devoted to geometry of Banach spaces; here one can find the principle of local reflexivity and Krivine’s and Dvoretzky’s (qualitative variant) theorems. The next chapter 12, An Introduction to Local Theory, deals with a quantative aspect of the Dvoretzky theorem (in particular, important Milman’s estimates), the Kadets-Snobar theorem about the \(\sqrt{n}\)-complementarity of \(n\)-dimensional subspaces of Banach spaces, and, finally, the Lindenstrauss-Tzafriri theorem on the characterization of a Hilbert space in terms of complemented subspaces. The last chapter 13, Important Examples of Banach Spaces, is devoted to some cumbersome examples of Banach spaces playing a fundamental role in understanding this excellent branch of Functional Analysis. The first 12 chapters contain special problem sections, presenting altogether 122 problems. The main part of them deals with additional material, related to the basic substance of the book, however, left out of the main body of the text. Among them are: the bounded approximation property, Orlicz sequence spaces, the Stone-Weierstrass theorem, the Amir-Cambern theorem, the James distortion theorem, superreflexivity, and many others.

At the end of the book, seven “supplements” are given. They contain fundamental definitions and results of linear functional analysis, headlined: Fundamental Notions, Elementary Hilbert Space Theory, Main Features of Finite-Dimensional Spaces, Cornerstone Theorems of Functional Analysis, Convex Sets and Extreme Points, The Weak Topology, Weak Compactness of Sets and Operators. The book also contains a List of Symbols, References, and an Index.

The book grew out of a one-semester course (in 2001) and subsequent two-semester courses (in 2004-2005) given by Nigel J. Kalton at the University of Missouri–Columbia. It is intended for graduate students and specialists in classical functional analysis. To my mind, this book is a good complement to other books devoted to geometry of Banach spaces, and I think that every mathematician who is interested in geometry of Banach spaces should, at least, look over this book. Undoubtedly, the book will be a useful addition to any mathematical library.

Reviewer: Peter Zabreiko (Minsk)