## AMS Mathematical Review 2001k:47001

by Robert M. Kauffman (Birmingham,AL),
E-mail:kauffman@math.uab.edu
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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