Barbara D. MacCluer’s compact text, Elementary Functional Analysis, is an unusual book on a number of counts. For one, it unmistakably conveys the author’s abundant enthusiasm for her subject as well as her evident joy at presenting mathematics in general: it’s a rare thing that a GTM text is riddled with apposite quotes, anecdotes, and historical asides, all making for a wonderful personal touch and drawing the reader into dialogue with the author in an almost palpable way. It works quite well and makes for an enhanced experience of learning this beautiful but occasionally austere material in an almost informal setting. Talking about flesh and blood mathematicians goes a long way toward taking the edge off, so to speak, as this intrinsically exciting but dense material is presented more palatably by virtue of introducing this human element. It makes for very good reading.
Each of the six chapters of the book in introduced by a well-chosen quote, often hinting in a very useful manner at the material that is to follow. I particularly like MacCluer’s choice of Dunford and Schwartz to start off her third chapter: “In linear spaces with a suitable topology one encounters three far-reaching principles concerning continuous linear transformations…” We find out quickly that these “Big Three” (as the chapter is titled) are uniform boundedness, the open mapping theorem, and Hahn-Banach. MacCluer quickly goes on to cover these three gems in a most effective and elegant manner, as well as a number of their corollaries or, in her words, “close cousins,” such as the closed graph theorem and Banach-Steinhaus.
And this brings me to the next count on which Elementary Functional Analysis is unusual: the density of gorgeous mathematics, presented elegantly and concisely, is truly remarkable, even if the text is meant only as an introduction to what the author herself calls a “huge field.” Everything is proved completely, at a pretty high but not uncomfortable pace, and, as already indicated, the flow of material is excellent. What’s more, each chapter is followed by a long list of exercises that should afford the reader the opportunity to take it all to the next level, from absorbing the presentation (take notes, fill the margin of the book with comments, doodle… whatever floats your boat…) to doing battle on one’s own, so to speak (although some problems come equipped with hints).
So it is that in the span of about 200 pages, MacCluer takes the reader from Hilbert space preliminaries to Banach- and C*-algebras and, of course, the spectral theorem (introduced by a quote from Paul Halmos stressing that it all begins with Hermitian matrices). The book’s preface indicates that MacCluer’s intent is to present an effective introduction to the “huge field” of functional analysis, suitable for self-study, with modest preliminary requirements placed on the reader. She achieves her goal beautifully: Elementary Functional Analysis is a wonderful book.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.