This is the ninth entry in the MAA’s *Guide* series, a subseries of the Dolciani Mathematical Expositions. The idea behind this series is to make topics in mathematics accessible to mathematically sophisticated non-specialists who are looking for an opportunity to get a quick overview of the subject. Graduate students studying for qualifying exams would be an obvious example of this cohort, and most or all of the books in this series (this one included) specifically mention that particular target audience. This series should also appeal to people whose student days are behind them but are not specialists in a given area, and who would like to either refresh their memory of a particular subject or get exposed to the main ideas quickly and efficiently.

This is an excellent set of books, and I am pleased to have had the opportunity to review the last four of them. They are all quite slim, and written by people who have both expertise in the subject matter and also a knack for succinct, elegant mathematical exposition. The ability to write like this is not insignificant; Blaise Pascal, after all, once famously apologized for writing a long letter by saying he didn’t have time to write a short one. Steven Krantz, the author of this book, has had a lot of practice in writing succinctly: in addition to his other books, this is his fourth entry in the *Guide* series (the others are on complex variables, real variables and topology).

This book (barely over a hundred pages of text) is very short, even by the standards of this series, but nevertheless addresses most or all of the standard topics that one would expect to see in an introductory graduate-level semester in functional analysis, and perhaps even one or two things that might not get mentioned. More specifically, the first chapter starts with normed linear spaces, then defines Banach spaces and discusses the “big three” results typically associated with them (Uniform Boundedness, Open Mapping, Hahn-Banach). This is followed by chapters on the dual space, Hilbert space, the algebra of bounded linear operators on a Banach space (including a fairly lengthy section on compact operators), and Banach algebras. The author then generalizes things by discussing (chapter 6) arbitrary topological vector spaces. The four remaining chapters of the text discuss, in order, distributions, spectral theory (for bounded, particularly bounded normal, operators on a Hilbert space; some background in measure theory is needed for this chapter), convexity (including the Krein-Milman theorem), and fixed point theorems (the contraction mapping principle and the Schauder theorem).

The author has taken pains to make the book accessible. Measure theory, as noted above, is used in the chapter on spectral theory and also in some examples, but other than that a good grounding in real analysis and linear algebra should get a student through most of the book. There is some inconsistency, however, in the expectations of the audience’s background: at one point, for example, the author feels the need to remind the reader of the definition of a metric space, but later in the book we read “We interpret derivatives in a Banach space in the usual Fréchet sense.” My guess is, however, that most of the people reading this book will already have been exposed to functional analysis and will be looking to refresh their memory rather than learn the material for the first time, so occasional statements like this should not prove troublesome.

Like other books in the *Guide* series, this one contains a good selection of examples, which I think is crucial. Another particularly nice feature of this book is the attention paid to applications of functional analysis, which even longer books frequently overlook. As some (non-exhaustive) examples, we see here, for example, the Uniform Boundedness theorem used to prove the existence of a broad class of functions with divergent Fourier series, the Hahn-Banach theorem invoked to establish the existence of the Green’s function for smoothly bounded domains in the plane, and the contraction mapping principle used both to establish an existence-uniqueness theorem for differential equations and to give an elegant proof of the implicit function theorem.

*Unlike* a number of other books in this series, however, this one also contains many proofs, even when they are decidedly non-trivial: proofs are given (or at least succinctly sketched), for example, of the Baire Category theorem, the “big three” results mentioned earlier, the spectral radius theorem for Banach algebras, the Schauder fixed point theorem, and other sophisticated results.

Opinions may certainly vary, but I am inclined to think that in very short books, particularly books in this *Guide* series, it is probably better to omit most or all proofs in favor of examples and broad, intuitive explanations of why something *should* be true. (The *Guide to Groups, Rings and Fields* refers to such explanations as “shadows of proofs”.) My feeling is that most people who want or need an actual proof in the first place will want to read one with the details spelled out, even at the expense of succinctness. Even excellent expositors cannot perform the impossible, and making proofs of difficult results completely comprehensible in a subject like functional analysis, in the space of a hundred pages or so, may well be asking for the impossible. Despite Krantz’s considerable skills in this area, there were times when, reading this book, I found the exposition a bit *too* concise for my taste.

I would have, for example, appreciated a bit more detail and a bit more motivation in the section on the spectral theorem; George Simmons, in his book *Introduction to Topology and Modern Analysis*, does not prove the integral version of the spectral theorem that is proved here, but does spend a page or two explaining how it is a generalization of the familiar result (for finite-dimensional normal operators) that one learns about in sophisticated linear algebra courses.

But, as I said, this is a subject on which people can reasonably disagree, and I am certainly not going to be critical of a person who attempts, even without complete success, to provide succinct proofs of theorems. If nothing else, such proofs give an overview of *how* the result is proved, even if all the details are not spelled out.

This book continues the tradition of high-quality expositions that have characterized every other Guide that I have looked at. This series in general, and this book in particular, deserve, and I hope will get, a wide audience. I also hope that I will be able to snag the tenth book in this series to review when it becomes available.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.