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Linear and Nonlinear Functional Analysis with Applications

Philippe G. Ciarlet
Publication Date: 
Number of Pages: 
[Reviewed by
Dhruba Adhikari
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This is a single-volume almost comprehensive textbook for linear and nonlinear functional analysts and/or those who are interested in the applications of linear and nonlinear analysis to partial differential equations. Selected topics from numerical analysis and optimization theory are also covered. The main focus throughout the volume has remained in a clear presentation of the most important and basic theorems and their applications. Numerous examples illustrating key topics and carefully selected applications of the theory are what make the textbook somewhat different from other textbooks with a similar intent.

Very cautiously selected exercises at the end of each section will undoubtedly support and enhance the understanding of the topics covered in that section. Some exercises are even theorems in some other books. The hints for some exercises are so great that important facts are very unlikely to be left out on a self-study of the book. The proofs of most theorems are self-contained and complete. The sources for important theorems whose proofs or presentations cannot be easily located in the literature are referenced along the way; usually at the bottom of the page. The bibliographical notes and references at the end of the book are so extensive that an enthusiastic reader will find them extremely useful.

A chapter on the “Great Theorems” of linear functional analysis includes Baire’s theorem and applications, the Banach-Steinhaus theorem and its several applications, the closed graph and open mapping theorems and applications, versions of the Hahn-Banach theorem, the Banach-Saks-Mazur theorem and the Banach-Eberlein-Smulian theorem.

An entire chapter is devoted to the fundamentals of Differential Geometry in \(\mathbb{R}^n\). The chapter also includes the existence and uniqueness theorems (the fundamental theorem of surface theory and the rigidity theorem for surfaces) for two highly nonlinear systems of partial differential equations.

There is also a chapter on the “Great Theorems” of nonlinear functional analysis. This chapter covers the most important and fundamental topics in nonlinear analysis, such as Euler-Lagrange equations, coercivity, the von Kármán equations, the existence of minimizers in Sobolev spaces, the p-Laplace operator, the Ekeland’s variational principle, applications of Brouwer’s fixed point theorems to the von Kármán equations and the Navier-Stokes equations by the Galerkin method, Lerary-Schauder theory, nonlinear monotone operators, and topological degree theory in finite dimensions.

A successful study of the book should lay a solid foundation for senior undergraduate and graduate students whose interests are in the application of the theory to linear and nonlinear partial differential equations. The book can also be an excellent reference for advanced researchers in the field of linear and nonlinear functional analysis.

Dhruba Adhikari is an assistant professor of mathematics at Southern Polytechnic State University, Marietta, Georgia.