This is an introduction to functional analysis at the early graduate level, with a strong emphasis on applications (primarily to differential and integral equations). Although this book is intended as a graduate text (and is too difficult for undergraduates at most American universities), it is noteworthy for how much background material is explicitly discussed in the text rather than merely assumed. For example, the first section of the book reviews basic linear algebra of finite-dimensional vector spaces. Even more surprising, some Appendix sections discuss such elementary matters as the union and intersection of sets and bijectivity of functions. Perhaps the reason for this is that the authors want this book to be accessible to graduate students in disciplines other than mathematics, such as physics or engineering.

There are ten chapters. The first, as just noted, begins with elementary linear algebra and also discusses basic metric spaces and normed linear spaces. Banach and Hilbert spaces are introduced as well, and there is a short section in which some standard inequalities (Young, Holder, Minkowski) are stated and proved. The next two chapters comprise a mini-course in measure theory. Measures and integration are discussed in the first of these chapters, and the Lp spaces are discussed in the second. The discussion in these chapters is fairly detailed and covers about 130 pages, roughly a third of the text.

Linear operators and linear functionals are discussed in chapters 4 and 5. The major theorems of functional analysis (Banach-Steinhaus, Uniform Boundedness, Hahn-Banach) are stated and proved. (The Hahn-Banach theorem is stated in its analytic form (extending functionals), but the geometric version (separating hyperplanes) is not discussed. The chapter on linear functionals also proves the Riesz representation theorem for functionals on a Hilbert space, but the result is not given that name, perhaps because it has been used previously for characterizing the functionals on the Lp spaces. Chapters 6 and 7 discuss some specific types of operators. The first of these chapters begins with some topological preliminaries on compactness in metric spaces, provides a statement and proof of the Arzela-Ascoli theorem, and culminates in a three-page discussion of compact operators. Then chapter 7 defines and gives basic results on various kinds of operators on a Hilbert space: self-adjoint, unitary and projection. Chapters 8 and 9 are concerned with the calculus of vector-valued functions of a real variable, and families of linear operators between Banach spaces that are smoothly parametrized. More applications to integral equations are given. The last chapter of the book, a fairly extensive one, discusses fixed point theorems and gives numerous applications to integral and differential equations.

The chapters generally have roughly the same structure. The text incorporates a number of examples and (sometimes) exercises embedded in the chapter, though this is somewhat hit and miss; especially in the later chapters, there are some that have only one or two embedded exercises and some have literally none. Then, a number of chapters end with a section titled “Advanced Practical Problems”; these are somewhat more demanding than the exercises that are sprinkled throughout the body of the text.

In this regard, it should be noted that the preface promises more than it delivers. The authors write that each chapter “consists of results with their detailed proofs, numerous examples, and exercises with solutions. Each chapter concludes with a section featuring advanced practical problems with solutions followed by a section on notes and references, explaining its context within the existing literature.“ However, except for one or two isolated cases, I found no solutions whatsoever either to the exercises or end-of-chapter advanced practical problems. While the lack of solutions is not troublesome (in fact, I think that easy access to solutions may have certain pedagogical disadvantages), I can’t help but wonder why the Preface says that they are there when they clearly are not.

The quoted statement above is also inaccurate in other respects. Not every chapter, as previously noted, contains exercises, and the sections on “advanced practical problems” disappear after chapter 6. Also, not one of the chapters has a “section on notes and references”; there is a bibliography at the end of the book, consisting almost entirely of journal articles, but no attempt is made to annotate this bibliography or explain, for each topic, how it fits within the framework of existing literature.

One respect in which the quoted statement is accurate is the assertion of “detailed proofs” of the theorems in this book. The authors (again, perhaps envisioning an audience consisting of graduate students from disciplines other than mathematics) do, indeed, write out things in detail; in chapter 4 on linear operators, for example, the book spends half a page proving the completely elementary result that the inverse of a bijective linear transformation is linear. Elsewhere (pages 54 – 60), the authors spend more than three full pages proving that a norm that satisfies the parallelogram law is induced by an inner product. Do graduate students really need this level of detail?

These kinds of detailed proofs will no doubt be appreciated by anybody who is attempting to use this book for self-study. On the other hand, the time and space spent on this level of detail must, of necessity, result in a trade-off, and many topics are not covered in the kind of depth that one might expect of a graduate text in functional analysis (at least for mathematics students). In fact, only the last three chapters (particularly chapter 10) engage in a reasonably deep investigation of the material covered in them. Throughout the rest of the text, one notices omissions that some might view as surprising for a book that purports to be a graduate text: compact, self-adjoint and unitary operators are defined, but not studied in any real depth; there is, for example, no discussion of spectral theory. The Baire Category Theorem does not appear, and neither do the Closed Graph and Open Mapping Theorems (though a standard consequence of the former, namely that the inverse of a bijective bounded linear transformation is also bounded, is proved). More sophisticated topics like Banach algebras, unbounded operators, topological vector spaces, and Sobolev spaces and distributions, are also omitted. The weak and weak* topologies are also not investigated in any depth. The authors do state, however, that a second (and perhaps a third) volume is envisioned, so presumably, they are saving some of these topics for subsequent books.

The authors’ writing style, though detailed, is often rather dry. This is not a chatty, conversational book. Reading it reminded me of the old TV show Dragnet and its “Just the facts, ma’am” mantra. Readers who like motivational discussions, therefore, may find themselves disappointed. It also appeared to me that some terminology is nonstandard. The authors use the phrase “abstract function” to refer to a function that I grew up calling a “vector valued function”, and defines weak convergence of linear functionals (page 244) where I would refer to “weak* convergence”.

To summarize and conclude: I doubt there is enough material here to enable this book to be used as a text for a graduate functional analysis course for mathematics students (particularly if, as is often the case at American universities, the course presupposes a prior course in measure-theoretic real analysis); better choices for an instructor looking for such a text might be Conway’s

A Course in Abstract Analysis, Sacks’

Techniques of Functional Analysis for Differential and Integral Equations, or

Functional Analysis, Spectral Theory and Applications, by Einsiedler and Ward. All three of these books have more extensive topic coverage than the book now under review and are therefore more suited for a graduate course for young mathematicians. However, the detailed proofs and extensive background material in this book likely make it useful for self-study or, possibly, as a text for a graduate course for non-mathematicians. The book may also be useful as a reference for instructors looking for good applications for a course in functional analysis, or who want a more-extensive-than-usual discussion of fixed point theorems.