What is functional analysis? If you sample a dozen textbooks with those words in the title, you might not be able to identify any items that uniquely characterize it. The Princeton Companion to Mathematics does not even identify it as a “branch of mathematics”. Yet most of us think of it as a real subject with a certain standard content. This would include at least a treatment of function spaces (usually Lp spaces) and operators on them. The level of sophistication varies widely from book to book. While functional analysis was once primarily the subject of a graduate course, there are now strong texts (such as Saxe’s Beginning Functional Analysis) that are aimed in part at advanced undergraduates.
The current work is probably an outlier even in this fractured functional analysis world. About half the book contains more or less standard topics from a real analysis text (such as Rudin’s Real and Complex Analysis). The other half focuses on topics of primary interest in partial differential equations and potential theory. Rather oddly, several of what one thinks of as core theorems of functional analysis (such as the Open Mapping and Closed Graph theorems) are absent.
Even the spectral theorem gets short shrift: about three pages, with a one-sentence introduction. (“Spectral theory allows one to diagonalize symmetric compact operators.”) This is not a chatty book.
The book begins by zipping through a treatment of the real numbers, metric spaces, continuity and convergence. Then, continuing to move quickly, we see the integral in Cauchy and Lebesgue forms, multiple integrals and change of variables. Banach and Hilbert spaces appear next in a chapter on normed linear spaces. This is followed by a chapter on weak convergence and duality in Hilbert and Banach spaces.
The author provides little or no explanation about his selection of topics or where he is going. There is a hint in the preface, but only that. It becomes a little clearer in the last three chapters. The final chapter is about applications to elliptic problems and to analytic and geometric inequalities. The big results of the last chapter are an isoperimetric inequality and the Faber-Krahn inequality. These are proved using exclusively methods of functional analysis. To get there, the previous two chapters address Sobolev spaces, the embedding of Sobolev spaces in other function spaces and capacity theory. A non-expert can follow the thread page by page and theorem by theorem, but the author does very little to light the way.
The book is likely to have a limited set of potential readers except for specialists particularly interested in elliptic problems.
Bill Satzer (firstname.lastname@example.org) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.