This is a Dover reprint of a book first published in England in 1973, intended, according to the preface, “for students in the final year of a British honours mathematics course or equivalent.” It covers the central core of functional analysis: normed, Banach and Hilbert spaces, continuous linear operators, functionals and dual spaces, compact operators (briefly), and of course the major theorems of the subject, including Hahn-Banach, Closed Graph, and Open Mapping. The spectral theorem for compact self-adjoint operators on a Hilbert space (two forms of which are given) constitutes the grand finale of the text.
On this side of the Pond, undergraduate functional analysis courses are rare, although there are certainly some books that would be suitable for such a course: Rynne and Youngson’s Linear Functional Analysis and Saxe’s Beginning Functional Analysis come immediately to mind, and I also recently discovered that Simmons’ Introduction to Topology and Modern Analysis, a very good book that I used as a student in the very early 1970s for two undergraduate courses (one a basic course in point-set topology using the first half of the book and the other a small seminar in introductory functional analysis using the second), is still in print, courtesy of Krieger Publishing Company.
I doubt, however, that Pryce’s book, though intended as an undergraduate text when originally written, would work well as a text for a current American undergraduate course: British courses are generally more sophisticated than American ones, and courses offered 40 years ago are more demanding than courses offered today. One has only to look at the list of topics that the author assumes as background (including not just a good understanding of metric space theory, advanced calculus and linear algebra, which one might expect, but also measure theory and integration) to see the problems inherent in trying to use this book to teach current undergraduates. Chapter 1 of the text constitutes a rapid review of this material, but I suspect a student who has not already been exposed to it — and that would obviously include the vast majority of undergraduate mathematics majors — would be at a considerable disadvantage.
Too sophisticated for undergraduates, this book might unfortunately also be viewed by many instructors of graduate functional analysis courses as not being sophisticated enough. In particular, there are a number of topics that an instructor of such a course might want to cover (including nonlinear theory, unbounded operators, general topological vector spaces, and distributions) that are not covered at all here; other topics, such as Banach algebras and spectral theory, are discussed here to some extent, but perhaps not as deeply as one would want to cover them in a graduate course. Other standard graduate functional analysis texts such as Rudin’s difficult but rewarding Functional Analysis (which may now be out of print, since it no longer appears on the McGraw-Hill webpage) or Lax’s Functional Analysis, provide much greater coverage than does this book. In other words, people who teach this subject well find this book (to borrow a phrase used by my parents to describe me as a teenager) “too light for heavy work and too heavy for light work.”
Despite my concerns about the lack of a natural constituency for this text, I certainly do not think it is a bad book. Quite the contrary: what this book does, it does very well. The writing is consistently crisp and clear, and the discussions are often very interesting and enlightening. I was particularly impressed, for example, with the text’s development of the Hahn-Banach theorem: the author discusses both its analytic (functional-extending) and geometric (separating hyperplanes) versions, and provides as an example of its use a proof of Runge’s approximation theorem in complex variables.
This last point should be emphasized: many introductory texts on functional analysis are usually so concerned with getting the basic material proved that they may not provide examples of non-trivial applications of the subject. (This was the “big gripe” noted in the review, in this column, of Rynne and Youngson’s book.) This book, by contrast, really strives to illustrate how functional analysis can be applied to “hard” analysis. In addition to the proof of Runge’s theorem, we have the following: in chapter 2, Ascoli’s Theorem is used to prove Montel’s theorem in complex analysis (some background in complex analysis is assumed here); Hilbert space theory is invoked in chapter 4 to discuss Fourier series and to prove the L2 convergence theorem and Fejér’s Theorem (three different proofs of which are given); chapter 7 contains a fairly extensive discussion of Sturm-Liouville theory, including the Green’s function, as an application of the theory of compact operators. These applications are developed in the text as soon as the necessary supporting material is introduced, thereby helping to motivate some of these ideas as they are taught. Additional examples are also provided in the exercises (of which there are a nice assortment, not accompanied by solutions). These illustrations of the applicability of functional analysis to hard analysis are, in my opinion, a very attractive feature of this book.
Another attractive feature of the book is its price (as I write this, roughly $17 at Amazon.com), which is a pleasant change from the $150–$200 routinely charged for many current mathematics texts, and which reaffirms my long-held view that Dover Publications does a real service to the mathematical community, both by saving deserving books from oblivion and also making them available at prices that even graduate students can afford. So, even if this book is not likely to be used as the primary text in a graduate course, it is certainly one that the students in such a course might want to look at for independent reading.
Mark Hunacek (firstname.lastname@example.org) teaches mathematics at Iowa State University.