AMS Mathematical Review 2001k:47001

by Robert M. Kauffman (Birmingham,AL),

Many universities teach a course or sequence in functional analysis which is oriented toward students who are interested in partial differential equations. This book is intended for such a course. However, it goes well beyond the level of many other books. For example, in addition to material oriented around the spectral theory of compact operators and applications, the book covers semigroup theory, including the Hille-Yosida theorem, an introduction to Sobolev spaces, and nonlinear evolution equations. It appears to be self-contained and rigorous. It even covers the Bochner integral, which is very useful in semigroup theory. It is not so easy to find a readable, short account of this material. This book seems to do the job. It is a suitable preparation for a more advanced course, at the level of, say, W. Rudin's book [Functional analysis, McGraw-Hill, New York, 1973; MR 51 #1315]. It also is a good background reference for a course at the Rudin level, if it is desired to include applications. I may use it that way myself, next year.

Many good examples are included, such as the harmonic oscillator and Bessel functions. Very useful but somewhat advanced material such as the theory of sectorial forms is also covered. One nice feature of the treatment of these advanced topics is that the aspects of them which are most important for applications are selected, then applied. The whole book is less than three hundred pages long.

I have not used it as a text. I tried it out on one of my Ph.D. students, who felt that it was quite readable. My feeling is that the material is very appropriate, the book is interesting, and that I may well use it in my teaching.

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