Functional Analysis: Theory and Applications

Functional analysis is an intrinsically beautiful subject and its applications are boundless. Its roots are at least in part in quantum mechanics, especially its formulation by von Neumann; it is integral to unitary representation theory; and it is central to many other branches of analysis, of both the hard and the soft kinds. Historically its origins go back at least to David Hilbert and Stefan Banach, and also, e.g., the Riesz brothers (Frygies and Marcel) and of course the aforementioned John von Neumann. Even Alexander Grothendieck famously worked in the field at the start of his career, before his conversion to algebraic geometry: quite a pedigree.

Thus, every mathematician needs to be versed in this subject to some degree: proper familiarity with the major results concerning Hilbert space, spectral theory, and so on. Our graduate courses are well-defined in this regard, and rightly so. There are numerous established texts on the subject, including the classics by Walter Rudin, and, for example, the books A Short Course on Spectral Theory, by William Arveson, and Elementary Functional Analysis, by Barbara MacCluer, which I have had the pleasure to review very positively in this column. While the two books by Rudin (i.e. his Real and Complex Analysis, a.k.a. “Green Rudin,” and Functional Analysis) are presumably still the unchallenged favorites in graduate schools from New Haven to La Jolla, and probably even beyond our borders, the other two books have marvelous qualities all their own to recommend them, particularly their compactness and foci commensurate with a perhaps more general objective: they seek to serve as launching pads for further studies in the subject at hand, while being essentially self-contained.

This said, the book under review at present, R. E. Edwards’ 1965 Functional Analysis: Theory and Applications, is in many ways suited as the indicated next step, to be taken after the novice has studied one or both of the books by MacCluer and Arveson (each of which is exceedingly accessible). Coming in at almost 800 pages, Edwards’ book is not the sort of thing one plows through linearly — well, it can of course serve that purpose, but the prospect is more than a bit daunting. Edwards’ book is encyclopedic and exceedingly thorough, so it can easily serve as a source not only for further (and more specialized) study beyond a first course, but also as something of a reference book for functional analysis, modulo the book’s age and the tastes of the author. On the other hand, it also a textbook, or rather a meta-textbook; Edwards appends exercise sets to each of his chapters, and they’re quite ramified.

In any case, to say that this book is thorough is a silly understatement: Edwards’ first (or 0th) chapter starts off with “Preliminaries from Set Theory” and ends with the theorems of Ascoli and Brouwer (i.e. the latter’s fixed point theorem), and the second (or first) chapter does linear algebra to such a degree that Hilbert spaces don’t make their appearance until page 89. But this is not to say that there isn’t a lot of functional analysis already knocking on the door earlier: this chapter explicitly features topological vector spaces and Hahn-Banach “in analytic form” appears on page 53. On the other hand, the rubber really hits the road in the next chapter, which is entirely devoted to the Hahn-Banach Theorem; and this business is then extended and expanded in the next chapter, where fixed point theorems are featured.

After this we get, in order, Radon measures (and dual spaces), distributions and PDEs, the open mapping theorem and the closed graph theorem, boundedness principles, duality, compact operators, and finally Krein-Milman. And there are a huge number of applications, asides, subthemes, and more, strewn though the 800 pages of the book, and a lot of it is pretty esoteric, even today, almost fifty years after the book’s appearance.

But this is all consistent with the aim of the work, i.e. serious scholarship at a high level presented for a large audience, factoring in Edwards’ own predilections. More than half the fun comes from the density of classic material, from Edwards’ scrupulous presentation of so much classical functional analysis, replete with proper attribution and in context. Accordingly the book under review sports 32 pages of bibliographic material. Indeed, within the limits it sets for itself, this is truly encyclopedic coverage. It is a very valuable book and is clearly aging well.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.