This well-crafted and scholarly book, intended as an (extremely) advanced undergraduate or early graduate text, scores on several fronts. For the well-prepared mathematics student it provides a solid introduction to functional analysis in the form of the theory of Banach spaces and algebras. It’s full of marvelously tantalizing preludes, in the sections marked “Notes” appearing at the ends of chapters, to what might come next in the development of the topics presented in the preceding discussions. It is clear as vodka that these notes are geared toward leading the student, or reader, in the direction of further work, and to do so quickly. For instance, the authors effectively add references to open problems fitted into the context of what is going on currently in the corresponding parts of functional analysis. Thus, the book proper provides fine coverage of the indicated functional analysis subjects pitched at a serious but by no means inaccessible level.
This is all to be expected in light of the success rate the late Graham Allan enjoyed at the University of Cambridge: just take a look at the book’s Preface, written by his former student Garth Dales. It is the latter who prepared the book from Allan’s TeX files and such, coming from his lecture over the years; Dales took up this task when Allan was taken seriously ill in 2006. This wonderful mitzvah panned out beautifully: the end result is an excellent book that promises to be of great use to a large audience.
The book is split into three parts: Banach spaces, Banach algebras, and the interface between Banach algebras and several complex variables. The prerequisites for reading the book are relatively modest; there is no requirement that the student should know anything about Lebesgue integration, and qua analysis proper, nothing is assumed beyond undergraduate level real variable theory (including metric space topology) and complex analysis: serious but not extraordinary stuff. Thus, this Introduction to Banach Spaces and Algebras necessarily covers a lot of general functional (and other) analysis under the Banach name, so to speak. For instance, “[o]ur account [of Banach algebra theory] … also describes the holomorphic functional calculus in one variable,” and the Daniell integral (in lieu of Lebesgue’s integral) is presented in connection with C*-algebras (or, more precisely, the Borel functional calculus, which is described as a generalization of the continuous functional calculus engendering the study of algebras with an involution: a C*-algebra is such a beast answering to what the authors call “the famous C* condition”: |a*a|=|a|2).
It bears mentioning, or reiterating, that the book under review does take the (presumably mathematically pretty young) reader to substantial heights, and he should therefore be prepared for such an adventure. Toward the end of the book, for instance, Dolbeault cohomology enters the scene, setting the stage for the Cousin problem (and, by the way, on pp. 327–328 there is a very elegant and compact rendering of the Dolbeault-Grothendieck Lemma).
Dales has been scrupulous about adding exercise sets to the presentation, so that the finished product really leaves nothing to be desired: this is a fine way to get into this beautiful subject and will serve to reel in a huge number of future devotees.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.