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Ruth Curtain, A. J. Pritchard, "Functional Analysis in Modern Applied Mathematics"

SIAM Review

This is an abstract book written in a modern, up to date style at about the second year graduate level, which seeks to bring the unifying principles of functional analysis to bear on diverse problems in applied mathematics. It is addressed to students and researchers in applied mathematics and the theoretical sciences.

The book is divided into three parts. Part I is entitled "Basic Functional Analysis" and in 112 pages takes up normed linear spaces, Hilbert spaces, measure and integration theory, probability theory, calculus in Banach spaces and general topological spaces and it mentions most of the classical theorems in those areas. The reader is assumed already to have had a good course in functional analysis and some exposure to probability theory, so that no proofs are given. The discussion is motivated by examples which illustrate and de-lineate the theory. However, the main purpose of Part I is to serve as a review of the topics and to set up the notation. Consequently, this part of the book has much of the character of a dictionary. In the event more detail is desired, references are given to sources where a more complete treatment can be found.

Part II is entitled "Analysis of abstract equations" and consists of two chapters, one on differential equations and one on spectral theory and applications. Awealth of topics are treated in a terse and concise manner including the abstract theory of ordinary and partial differential equations, distribution theory, semi-group theory, evolution equations, spectral theory for compact normal operators and the spectral decomposition of unbounded linear operators. The proofs of most results are given. However, a working knowledge of ordinary and partial differential equations would be useful. Finally, part III takes up a variety of applications and is in my opinion the nicest part of the book. Proofs are given in detail. There are numerous examples, many of them nontrivial, which lead to a deeper understanding of the material and motivate the development. The point of the chapter is to see what one can say, using general functional analytic techniques about applied problems. Everything is kept in a general setting. Topics in stability theory, linear systems theory, optimization and optimal control theory, numerical analysis and infinite dimensional linear systems theory are discussed. Consistent with the goals of the book, numerical methods, for example, are analyzed but no computing is done.

From the tenor of the above remarks, one is tempted to conclude that the book is of limited usefulness to the practitioner, and due to a lack of problems, does not lend itself either to self-study or to use as a text book in a course. That view is too superficial. First, precisely in applied areas, more and more is being written in the language of functional analysis and this book gives insights into both the language and what one can expect from functional analysis. In fact, this was one of the motivating factors which prompted the authors to write this book. Secondly, if one really studied this book, followed up the intriguing questions which are often raised, and delved further into the various topics by following up some of the numerous references, several excellent courses based on this book could be given. This is a nice book. Great care was taken in selecting the material and the examples. The layout is attractive and the proof reading was conscientous. The authors have succeeded very well in accomplishing the goals they have set for themselves.

--RONALD B. GUENTHER, Hahn-Meitner Institute, Berlin

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