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Linear and nonlinear functional analysis with applications. With 401 problems and 52 figures. (English) Zbl 1293.46001
Other Titles in Applied Mathematics 130. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). (ISBN 978-1-611972-58-0/hbk). xiv, 832 p. (2013).
Functional analysis evolved as a natural gathering point for a number of investigations regarding the solvability of many classes of problems, including ordinary and partial differential equations, or linear equations which are either in the form of integral equations or in the form of countable systems of linear scalar equations in which the unknown is a sequence of numbers. As the subject developed, much broader areas of applicability became evident. These applications, in turn, spawned further abstract development, and the abstract results themselves assumed an intrinsic interest.
The author’s aim is to give a systematic treatment of some of the fundamental abstract results in linear and nonlinear functional analysis. They are illustrated by numerous relevant applications to certain concrete problems in linear and nonlinear partial differential equations, numerical analysis, and optimization theory. Moreover, by interlacing extensive commentary and foreshadowing subsequent developments within the formal scheme of statements and proofs, by the inclusion of many examples and by appending interesting and challenging exercises, the author has written a monumental book which is eminently suitable as a text for a graduate course. The author’s approach (as opposed to the traditional “theorem followed by proof” approach) makes the book more readable and therefore more attractive to students. This volume is distinguished by the broad variety of problems covered and by the abstract results developed.
The content of the book is divided into 9 chapters, as follows: 1. Real analysis and theory of functions: a quick review; 2. Normed vector spaces; 3. Banach spaces; 4. Inner-product spaces and Hilbert spaces; 5. The “great theorems” of linear functional analysis; 6. Linear partial differential equations; 7. Differential calculus in normed vector spaces; 8. Differential geometry in $${\mathbb R}^n$$; 9. The “great theorems” of nonlinear functional analysis. The volume contains 401 problems and 52 figures, plus historical notes and many original references that provide an idea of the genesis of the important results, and it covers most of the core topics from functional analysis. The list of references covers almost thirty pages (and is far from being complete!), and a list of symbols and an index will help the reader in effectively using this monumental reference book whose presentation is excellent.
In summary, this substantial volume is a tour de force by the author, who is a master of applied nonlinear functional analysis and who has contributed extensively to the development of the theory of partial differential equations. The volume under review deals rigorously with mathematical models of high applicability to real world problems. From this viewpoint it is a significant contribution to a currently active area of research.
Globally, the reviewer very much likes the spirit and the scope of the book. The writing is lively, the material is diverse and maintains a strong unity. The interplay between the abstract functional analysis and relevant concrete problems arising in applications is emphasized throughout. On balance, the book is a very useful contribution to the growing literature on this circle of ideas. I wholeheartedly recommend this book both as a textbook, as well as for independent study. In my opinion, this book is a reference work in the existing literature on nonlinear analysis and will remain a reference for many years.

##### MSC:
 46-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis 47-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operator theory 35-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations 46N20 Applications of functional analysis to differential and integral equations 47N20 Applications of operator theory to differential and integral equations