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Introduction to Banach Spaces: Analysis and Probability, Volumes 1 and 2

Daniel Li and Hervé Queffélec
Cambridge University Press
Publication Date: 
Cambridge Studies in Advanced Mathematics 166/167
[Reviewed by
Allen Stenger
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This two-volume work gives a comprehensive overview of Banach spaces that is slanted toward probability theory. It was published as a single volume in French in 2004, and in English translation in two volumes in 2018. There’s not a sharp break between the volumes, but the first volume concentrates on tools and the second volume on properties of Banach spaces. The content has not been updated for this translation, except for some clarifications and corrections, but there are four brief appendices by guest scholars that survey results of the past 20 years. The chapter endnotes are especially good and show off the authors’ vast knowledge of the subject. There are also reasonable sets of exercises.

The book assumes the reader is already familiar with functional analysis, abstract spaces, and general measure theory. The authors recommend the first ten chapters of Rudin’s Real and Complex Analysis as a prerequisite. The book sticks mostly to the general theory of Banach spaces and their operators, and does not deal with any special kinds of spaces, such as \(H^p\)  spaces or spaces of analytic functions. It also does not deal with more specialized structures such as Banach algebras (except in an appendix) or Hilbert spaces. The book does try to show the many applications of the material to probability theory and (to a lesser extent) to harmonic analysis.

The layout of the book makes it easy to use. There’s a lengthy preface that gives detailed summaries of each section of the book. The front and end matter (except the preface and bibliography) is prepared separately for each volume, but then reprinted in both volumes. I think the reprinting is a good feature, but having two indices at the end of a volume is confusing and I would have liked a combined index better. Each volume is paginated separately.

The book is intended as a textbook, but at 840 pages it probably contains more information about Banach spaces than any student wants to know. I think it works better as a reference.

This is more advanced than most books on Banach spaces. A well-regarded text at a more elementary level, that I have not seen, is Allan’s Introduction to Banach Spaces and Algebras. Another book that is more of a reference but does not have quite the breadth of this one is Fabian et al.’s Banach Space Theory: The Basis for Linear and Nonlinear Analysis.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is His mathematical interests are number theory and classical analysis.

Volume 1: Preface
Preliminary chapter
1. Fundamental notions of probability
2. Bases in Banach spaces
3. Unconditional convergence
4. Banach space valued random variables
5. Type and cotype of Banach spaces. Factorisation through a Hilbert space
6. p-summing operators. Applications
7. Some properties of Lp-spaces
8. The Space l1
Annex. Banach algebras, compact abelian groups
Author index
Notation index
Subject index.
Volume 2: Preface
1. Euclidean sections
2. Separable Banach spaces without the approximation property
3. Gaussian processes
4. Reflexive subspaces of L1
5. The method of selectors. Examples of its use
6. The Pisier space of almost surely continuous functions. Applications
Appendix. News in the theory of infinite-dimensional Banach spaces in the past twenty years G. Godefroy
An update on some problems in high dimensional convex geometry and related probabilistic results O. Guédon
A few updates and pointers G. Pisier
On the mesh condition for Sidon sets L. Rodriguez-Piazza
Author index
Notation index
Subject index.