CMS Books in Mathematics. Berlin: Springer (ISBN 978-1-4419-7514-0/hbk; 978-1-4419-7515-7/ebook). xiii, 820 p. EUR 99.95/net; £ 90.50; SFR 143.50; $ 129.00 (2011).

The book under review is a substantial text (over 800 pages long) that combines an introduction to the basic principles of functional analysis with more advanced topics that lead the reader to many frontiers of current research in Banach spaces. From the Preface of the book: “ … It is the purpose of this introductory text to help the reader grasp the basic principles of Banach space theory and nonlinear geometric analysis. The text presents the basic principles and techniques that form the core of the theory. It is organized to help the reader proceed from the elementary part of the subject to more recent developments. … Experience shows that working through a large number of exercises, provided with hints that direct the reader, is one of the most efficient ways to master the subject. Exercises are of several levels of difficulty, ranging from simple exercises to important results or examples. They illustrate delicate points in the theory and introduce the reader to additional lines of research. … An effort has been made to ensure that the book can serve experts in related fields such as Optimization, Partial Differential Equations, Fixed Point Theory, Real Analysis, Topology, and Applied Mathematics, among others. As a prerequisite, basic undergraduate courses in calculus, linear algebra, and general topology, should suffice.” For the most part, the authors’ words from the Preface are accurate, although a student (at least, a student at most U.S. universities) who only has the “basic undergraduate courses in calculus, linear algebra, and general topology” will probably find portions of this book difficult to read. In the reviewer’s opinion, a course introducing the reader to measure theory is an advisable prerequisite to read this book. With this additional prerequisite, the book has much to recommend it as an introductory text. The book is well-written and is essentially self-contained. All of the standard topics (as well as many other topics) are covered and the authors have accumulated a large collection of exercises on which students can hone their skills. The exercises range from exercises that are easy and straightforward to exercises that are quite challenging and for which the reader will be thankful for the authors’ hints. The extensive collection of exercises with hints is something that this book has in common with the authors’ previous book [Functional analysis and infinite-dimensional geometry. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC 8. New York, NY: Springer (2001;

Zbl 0981.46001)] written with {\it J. Pelant} and is one of the features that distinguishes this book from some of the other introductory texts. The inclusion of remarks and open problems at the end of almost every chapter of this book alerts the reader to the dynamic nature of the field and leads the reader to understandable research questions. For a reader who has already been introduced to the basics of functional analysis, this book also provides a wealth of opportunities. The authors have included chapters that introduce the reader to many advanced topics in the theory of Banach spaces. For example, Chapter 5 on the structure of Banach spaces includes the theorem of Banach and Mazur that every separable Banach space embeds isometrically into $C[0,1]$, Sobczyk’s theorem on the complementability of $c_0$ in separable Banach spaces containing $c_0$, Pełczyński’s characterization of separable spaces containing $\ell_1$, Rosenthal’s $\ell_1$ theorem, as well as the $\ell_1$ theorems of Odell and Rosenthal and the theorem of Bourgain, Fremlin, and Talagrand concerning Baire-$1$ functions on Polish spaces. Chapter 5 ends with twelve remarks pointing the reader to related results, six open problems, and ninety-nine exercises. Other chapters introduce the reader to topics in the local theory of Banach spaces, the geometry of Banach spaces (including superreflexivity, dentability and the Radon-Nikodým property, and various hierarchies of smooth norms), optimization, fixed point theorems and nonlinear geometric analysis, weakly compactly generated Banach spaces and topics in the weak topologies on Banach spaces, compact operators on Banach spaces including spectral theory, and tensor products of Banach spaces, including Enflo’s example of a Banach space without the approximation property. Thus the book can be used for many different advanced course, depending on the instructor’s preferences. For the more advanced students and for researchers, the remarks and open problems are especially useful. The authors do not indicate what sections of the book they envision to comprise a one-semester graduate course, a two-semester graduate course, etc. This is not a problem for a course with an instructor, but it could be problematic for a student reading the book independently. Even some of the earlier chapters include topics that are not normally covered in an introductory one-semester course. For example, in Chapter 3, entitled Weak Topologies and Banach Spaces, standard topics such as the definitions of the weak and weak* topologies, weak compactness, reflexivity, the Banach-Steinhaus uniform boundedness principle, Alaoglu’s theorem, the Eberlein-Šmulian theorem, and the Krein-Milman theorem are found as well as less-standard topics such as Choquet’s representation theorem, spaces of distributions, and a proof of James’s characterization of reflexive spaces in terms of norm-attaining functionals. For a student beginning to learn the subject independently, the book may actually have too much material and such a student may find other books (including the authors’ book with Pelant mentioned earlier) easier to get through. In summary, the authors have written an impressive book that should be welcomed by students interested in learning the basic or more advanced topics in the theory of Banach spaces and by researchers in Banach spaces or related fields.