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Lecture notes on functional analysis. With applications to linear partial differential equations. (English) Zbl 1268.46001

Graduate Studies in Mathematics 143. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-8771-4/hbk). xii, 250 p. (2013).
This textbook grew out of a course that the author taught in 2011 at the Pennsylvania State University, USA. Accordingly, its main goal is to provide graduate students with a first introduction to the basic concepts, methods, and applications of functional analysis. As the author points out in the preface, this textbook is addressed to a wide audience of students in mathematics and other disciplines at the first-year graduate level, and most of its content can be covered in a one-semester course for such students. In view of this particular didactic purpose of the book, the following principles have been chosen to determine its general conception:
(1) The text has been kept as concise as possible, discussing all the fundamental concepts and results necessary for beginners, but avoiding any digressions otherwise.
(2) Throughout the book, a high value has been set on a careful explanation of the relations between theorems in functional analysis and familiar results in finite-dimensional linear algebra, thereby building on the background knowledge from the students’ undergraduate studies.
(3) Both the theory of Sobolev spaces and the topic of semigroups of linear operators on Banach spaces are developed far enough in order to present significant applications of functional-analytic methods to partial differential equations.
(4) Illustrations of the theoretical main ideas with figures have been included, whenever appropriate and possible, and an abundance of related homework problems has been added at the end of each chapter.
As for the precise contents, the book consists of nine chapters, each of which is divided in several sections.
Chapter 1, the introduction to the present book, highlights some of the key results of finite-dimensional linear algebra and their counterparts in (infinite-dimensional) linear functional analysis. This is done in an informal way, thereby touching upon linear partial differential equations, differential operators, evolution equations, function spaces, and further concepts explained in the course of the book. Chapter 2 treats the basics on Banach spaces and their linear operators, including seminorms and Fréchet spaces, the classical extension theorems for linear functionals, the separation of convex sets, dual spaces of Banach spaces, and the notions of weak and weak-star convergence in (separable) Banach spaces. Chapter 3 discusses spaces of continuous functions, with particular emphasis on spaces of bounded continuous functions, the Stone-Weierstrass approximation theorem, Ascoli’s compactness theorem, and spaces of Hölder continuous functions. In Chapter 4, bounded linear operators in Banach spaces are analysed more closely. The main topics of this chapter are the Banach-Steinhaus uniform boundedness principle, the open mapping theorem, the closed graph theorem, adjoint operators and their properties, compact operators and their adjoints, and – as an instructive example of compact operators – integral operators on the space \(C([a,b])\) of continuous, real-valued functions on the closed interval \([a,b]\).
Chapter 5 turns to the basic facts on Hilbert spaces and their linear functionals, including the Riesz representation theorem, the Gram-Schmidt orthogonalization algorithm, orthonormal bases, the example of complex Fourier series, positive-definite operators on Hilbert spaces and the Lax-Milgram theorem, and weakly convergent sequences in Hilbert spaces. Compact operators on a Hilbert space are described in Chapter 6, where the basics of Fredholm theory, spectra of compact operators, self-adjoint operators, and the Hilbert-Schmidt theorem (on the existence of a countable, orthonormal eigenbasis of a separable real Hilbert space with respect to a compact symmetric linear operator on it) represent the focal points of the discussion of this topic. Chapter 7 gives an introduction to the theory of semigroups of linear operators. Starting from ordinary differential equations in a Banach space, the definition of the classical matrix exponential function is extended to the concept of semigroups of linear operators on a Banach space. In this context, the connection with finite-dimensional ordinary differential equations is particularly stressed, the close relation between resolvent operators and “backward Euler approximations” is carefully described, and the problem of the existence and uniqueness of the semigroup generated by a given linear operator is discussed in great detail. Thereafter, Chapter 8 presents the basic theory of Sobolev spaces, especially as a useful abstract framework for later applications to partial differential equations. After an introduction to the theory of distributions and weak derivatives of locally summable functions, Sobolev spaces and their basic properties are explained in the sequel, together with a number of instructive examples. This chapter also contains more advanced material on approximations of Sobolev functions, relations between weak and strong derivatives of Sobolev functions, and various embedding theorems for Sobolev spaces. Furthermore, the reader will get acquainted with several classical inequalities for Sobolev functions, among which are those named after Morrey, Gagliardo-Nirenberg, Poincaré, and others.
Finally, the goal of Chapter 9 is to demonstrate how the abstract techniques of functional analysis can be applied to the theory of elliptic, parabolic, and hyperbolic partial differential equations. In the first section, the boundary value problem for special linear elliptic equations is treated in a weak form, involving bounded linear operators on a Hilbert-Sobolev space. It is then shown how unique solutions to this new equation can be obtained using functional-analytic methods such as Fredholm theory, self-adjoint operators, and series of orthogonal eigenfunctions. In the following two sections of this final chapter, initial value problems for evolution equations of parabolic and hyperbolic type, respectively, are studied by applying the theory of semigroups of linear operators on suitable function spaces. The last part of the book is an appendix, in which some background material is collected for the convenience of the reader. In the seven brief sections of this appendix, the reader finds a number of basic definitions and facts from general topology, Lebesgue measure theory, and real analysis, together with some of the most important inequalities used in the course of the book.
Each chapter ends with a list of related, very carefully selected exercises. These homework problems complement the main text in a very effective manner, thereby providing a wealth of additional examples and more advanced results. As the author points out, a complete set of solutions to the nearly 180 exercises is available to instructors upon request.
Altogether, this introduction to the fundamental concepts and methods of linear functional analysis stands out by its special didactic conception, on the one hand, and by its very lucid user-friendly presentation on the other. The book can be used as a valuable source for a one-semester graduate course on the subject, and it is just as suitable for profound self-study by interested students or working scientists. No doubt, this primer is a highly welcome addition to the existing textbook literature on functional analysis and its applications.

MSC:

46-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis
46Bxx Normed linear spaces and Banach spaces; Banach lattices
46Exx Linear function spaces and their duals
46Cxx Inner product spaces and their generalizations, Hilbert spaces
47D03 Groups and semigroups of linear operators
35-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations
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