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Linear functional analysis. (English) Zbl 1203.46001

Graduate Studies in Mathematics 116. Providence, RI: American Mathematical Society (AMS); Madrid: Real Sociedad Matemática Española (ISBN 978-0-8218-5115-9/hbk). xiii, 330 p. (2010).
This book is a co-publication of the American Mathematical Society and the Real Sociedad Matemática Española. The presentation is intended to be accessible to readers whose background includes basic linear algebra, integration theory, and general topology. The illustrative examples and carefully chosen exercises make it definitely a very attractive source for anyone learning or teaching functional analysis.
The book presents some basic facts of the linear functional analysis related to fundamental aspects of mathematical analysis and their applications. The content is divided into nine chapters, as follows:
1. Introduction; 2. Normed spaces and operators; 3. Fréchet spaces and Banach theorems; 4. Duality; 5. Weak topologies; 6. Distributions; 7. Fourier transform and Sobolev spaces; 8. Banach algebras; 9. Unbounded operators in Hilbert space. Almost 240 exercises are useful to the reader in better understanding the concepts employed, and hints to exercises are provided in the final part of the volume.
The material in the first part of the book can be used for an introductory course on functional analysis, with an emphasis on the role of duality. The second part introduces distributions and Sobolev spaces and their applications. Convolution and the Fourier transform are shown to be useful tools for the study of partial differential equations. Fundamental solutions and Green’s functions are considered and the theory is illustrated with several applications. In the last chapters, the Gelfand transform for Banach algebras is used to present the spectral theory of bounded and unbounded operators, which is then used in an introduction to the basic axioms of quantum mechanics.
In conclusion, the book is well written and is a pleasure to read. The author not only loves mathematics, but enjoys putting it into words. The reviewer warmly recommends this volume to graduate students interested in functional analysis, partial differential equations, and analysis.

MSC:

46-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis
46Axx Topological linear spaces and related structures
46Exx Linear function spaces and their duals
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
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