This is a text for a first course in functional analysis. Unlike some books (such as those by Saxe (*Beginning Functional Analysis*), Rynne and Youngson (*Linear Functional Analysis*), and Muscat (*Functional Analysis: An Introduction to Metric Spaces, Hilbert Spaces and Banach Algebras*), as well as the more novel approach taken by Shapiro in *Volterra Adventures*) that attempt to make the subject accessible to undergraduates, this text is, in terms of prerequisites, style of writing, topic coverage, and exercises, very definitely intended for graduate students, particularly those in their second year. For courses at that level, however, this is an attractive text.

The topics covered, and the order in which they are presented, are interesting and a little unusual. Instead of devoting separate chapters to Banach spaces, Hilbert spaces, and Banach algebras, this text introduces all three of these objects more or less simultaneously in the first chapter. So, within the first forty pages or so of text, the reader has seen the definitions of these spaces, the major examples of them, and some basic theorems concerning them, as well as an introduction to bounded linear operators and the dual space. The Baire Category theorem is also proved in this chapter, but applications of it are relegated to the exercises. In the next chapter, the “big three” results of functional analysis (Hahn-Banach, Open Mapping and Uniform Boundedness theorems) are proved. The Hahn-Banach theorem is discussed from both an analytic (extending linear functionals) and geometric (separating hyperplanes) point of view. James’s space, not often mentioned in the textbook literature, is given an extensive discussion here. (This is a Banach space that is not reflexive, but is isometrically isomorphic to its double-dual space.)

Chapter 3 introduces a more general structure, topological vector spaces, and focuses on issues relating to the weak and weak* topologies, such as the Banach-Alaoglu theorem and the Eberlein-Smulyan theorem. The Krein-Millman theorem is proved, and ergodic theory is introduced as an application of the material of this chapter.

Chapter 4 returns to Banach spaces and discusses compact operators and Fredholm theory. Integral equations are sometimes discussed as an application of these ideas (see, for example, *Functional Analysis, Spectral Theory and Applications* by Einsiedler and Ward, but are not here, although they are alluded to in the exercises.

Spectral theory is the subject of chapter 5, and various aspects of it are discussed. Before tackling the general theory, the book looks at compact operators on a Banach space. Then, after developing the holomorphic functional calculus, the text turns to bounded operators on Hilbert spaces, particularly self-adjoint and normal ones.

The final two chapters of the book cover what I might call “special topics,” issues not always covered in a first course on the subject. There is a chapter on unbounded operators and one on continuous semigroups of operators.

As the summary above illustrates, there is quite a lot of material covered in this text, certainly enough for a full year course. Because instructors using this text for a one-semester course in functional analysis will of course have to pick and choose what material to cover, a chapter dependence chart would have been helpful. While not providing that, the authors do discuss what was covered in a lecture course offered by Salamon in the fall 2015 semester; based on the description, it seems to have been an extensive course, with significant portions of each chapter of the book (except the one on unbounded operators) covered. My guess is that at most American universities, this much material could not be covered in a single semester.

In order to keep the size of the book within manageable limits, some topics must, inevitably, be omitted. The theory of distributions, which is sometimes discussed in graduate-level texts, is not covered in this text. Neither is nonlinear functional analysis. Specific applications of functional analysis (such as to differential equations) are sometimes alluded to in the text or in the exercises, but not developed in any great detail.

The book assumes a substantial mathematical background on the part of the reader. Familiarity with measure theory is necessary; the \(\mathcal{L}^p\) spaces, for example, are defined very early on (page 3, in fact) without preliminary definition of the word “measure”. (There is also no appendix covering this background.) Likewise, the reader is assumed to be familiar with the basics of real and complex analysis, linear algebra, and point-set topology. Phrases like “metric space” and “topology” are defined in the text, but at a pace that clearly assumes the reader has seen these terms before. An appendix proves (using the Bourbaki-Witt fixed point theorem) the equivalence of Zorn’s Lemma to the Axiom of Choice, and also offers a proof of Tychonoff’s theorem.

The authors both teach in Europe (ETH Zurich) and the book is, accordingly, written in a concise European style: things that need to be said, *are* said, but there is very little hand-holding. Students brought up on books with a very conversational, “chatty” tone might find the exposition here a bit dry: theorems and proofs and examples without very much in the way of conversational “filler.” On the other hand, there is certainly something to be said for teaching graduate students how to read serious mathematics.

Every chapter ends with a section of exercises, and, in addition, there are a number of exercises also embedded in the text itself. Few if any of the exercises are trivial, and a number of them call for proofs of well-known results. For example, in chapter 1, the reader is asked to prove, as a consequence of the Baire Category Theorem, that there exists a continuous, nowhere-differentiable function. Solutions to the exercises are not provided in the text, and there is apparently no solutions manual for instructors.

There is a reasonably extensive (more than four pages of small type) bibliography at the end of the book. The vast majority of the 88 references in this bibliography are to papers and articles rather than textbooks.

Conclusion: This is a demanding book, but a valuable one. Instructors teaching a course that, in the past, might have been based on a text like Rudin’s (now apparently out of print) *Functional Analysis* might well find this book a suitable alternative. Those looking for a book that is not so demanding, and therefore suitable for a first-year graduate course, might look at, for example, Conway’s *A Course in Abstract Analysis* or (especially for people wanting an applications-oriented text) Sacks’ *Techniques of Functional Analysis for Differential and Integral Equations.*

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Mark Hunacek ([email protected]) teaches mathematics at Iowa State University.