BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY

A functional analysis course is taught in almost all mathematics departments with a Ph.D. program. Functional analysis is primarily the study of the algebraic, analytic and geometric structures of infinite-dimensional vector spaces and operators on them. The prototypical example is spaces of functions, regarded as vector spaces, where the operators are differential and integral operators. The study of infinite dimensional spaces evolved naturally and eventually became a very rich, broad and sophisticated subject. At one time, it was difficult to find a good text book of functional analysis decades ago, but nowadays we have plenty of choices. Before having Lax’s book, we saw some very successful textbooks of functional analysis; to name a few: Yosida’s [10] and Conway’s [2]. Lax’s Functional Analysis, which grew out of a course taught by the author at the Courant Institute over many years, can be predicted to be another successful book. It can be easily adapted as a textbook for an introductory course, and it can be used as an excellent reference book for current and future analysts. The instructor can easily choose topics from the first twenty five chapters to teach a one-year course of introductory functional analysis. One should not be intimidated by the number of chapters: each chapter might be deliberately kept short. While the material covered in these chapters is quite similar to other books, the author makes a very thoughtful arrangement of the order: there are many chapters on theory followed immediately by chapters titled “Applications of” or “Examples of”. This reflects his intention to embed abstractions into real applications. The book starts by recalling definitions of linear spaces and linear maps in the first two chapters. Some simple properties about convex sets are given. Further study of these properties is seen in the discussion of locally convex spaces in chapter thirteen. Some nonstandard items about the index of linear maps (e.g. the product formula for the index) are also given in the first two chapters. The discussions of various types of Hahn-Banach theorems and their extensions are given in chapter three, which includes the theorem of Agnew and Morse on a semigroup (this is seldom seen in other textbooks) and the Hahn-Banach type theorem of Bohnenblust, Sobczyk and Soukhomlinoff for complex linear spaces. The next chapter is devoted to a variety of applications of Hahn-Banach theorems and the first historical note, the tragic death of Hausdorff. It returns to normed linear spaces in chapter five. Examples of Banach spaces, which include Sobolev spaces, are given. The striking difference between a finite dimensional ball and an infinite dimensional ball is emphasized and put in one section. Already, readers may observe the elegant structure of the book. Each section is relatively short and thus easy to digest. Exercises are offered in the middle of a section, not at the end, which surely is a natural way for people to read a book or to give lectures. Hilbert space is introduced in chapter six. The optimal distance from a given point to a closed nonempty convex set is discussed. This naturally leads to the