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Applied functional analysis. Main principles and their applications. (English) Zbl 0834.46003
Applied Mathematical Sciences. 109. New York, NY: Springer-Verlag. xvi, 404 p. (1995).
As the author states in the Preface, there are two ways of teaching mathematics, namely the systematic way (governed by the desire for mathematical perfection and completeness of the results), and the application-oriented way (motivated by the search for a possibly large variety of nontrivial applications). The two books under review are based, as their common title suggests, on the second approach. The author seems to have felt the necessity of collecting and comparing the basic principles of both linear and nonlinear analysis, with a particular emphasis on applications in mathematical physics. As a matter of fact, in many textbooks on functional analysis one can find such applications, if there are any, merely as a by-product or a collection of exercises.
Let us briefly describe the contents of both volumes. Chapter 1 of Volume 108 starts with some important tools of linear analysis (linear spaces and operators, dual spaces, Neumann series, spectra) and nonlinear analysis (Brouwer’s and Schauder’s fixed point theorems, the Banach- Caccioppoli theorem, the Leray-Schauder principle). The famous Dirichlet principle stands at the beginning of Chapter 2 which is devoted to Hilbert space theory. Here the notion of orthogonality is of course crucial, but also applications to boundary value problems, finite elements, and iteration-projection methods are discussed, and Minty’s existence and uniqueness principle for monotone operators is sketched.
Chapter 3 is concerned with the general problem of expansions of functions into series of eigenfunctions; this has its classical roots, of course, in the theory of Fourier series and integrals. In the following Chapter 4, the author gives a brief account of the (linear) Fredholm alternative and the Hilbert-Schmidt theory of eigenvalues of compact symmetric operators in Hilbert space; he also illustrates the abstract results by means of the usual integral equations and boundary value problems.
The last Chapter 5 with the heading “Selfadjoint operators, the Friedrichs extension and the partial differential equations of mathematical physics” may be considered as the core of the book. The reader may find here not only a lot of advanced functional analysis (e.g., the Friedrichs extension, semigroups of operators, trace class operators, \(C^*\) algebras), but also quite an impressive amount of applications to quantum mechanics and theoretical physics (e.g., the classical partial differential equations of mathematical physics, basic problems of quantum mechanics and quantum statistics, scattering theory, Feynman’s path integrals).
The second book under review (Vol. 109) constitutes a somewhat strange mixture of linear and nonlinear functional analysis. Chapter 1 deals with the Hahn-Banach theorem and its application to convex optimization problems. In Chapter 2 the author discusses variational methods, both linear and nonlinear. A basic device in the nonlinear theory is of course to look for a potential for a given operator satisfying a Palais-Smale condition, in order to apply the mountain pass lemma or its various generalizations. Chapter 3 discusses what is usually called the fundamental principles of linear functional analysis (open mapping, closed graph etc.) which all build on Baire’s category theorem. A central tool of nonlinear analysis in turn, namely the implicit function theorem, is described in detail in Chapter 4. The final Chapter 5 is concerned with Fredholm operators, both linear and nonlinear. Here the nonlinear part also covers some bifurcation theory with applications to nonlinear equations.
Without any doubt, both voumes constitute an excellent and useful enrichment of the existing vast literature on functional analysis, operator theory, and mathematical physics. The author has the talent to express even complicated facts in a simple and suggestive language (how many books and papers are written just the other way round!); as a consequence, the book may be warmly recommended also to non-specialists. Moreover, even students with a basic knowledge of calculus may find the books readable and interesting; so it is to be hoped that a cheaper paperback edition will be available soon.
Of course, there is an unavoidable nonempty set of misprints and oversights. For example, the author stresses explicitly the importance of Heisenberg’s uncertainty principle in the Preface of Vol. 108, but failed to notice the error in its precise formulation on p. 343. The reviewer also found the author’s predilection for “lists of everything” more confusing than illuminating, but this did not really diminish the pleasure when “reading” the two books.
Coming back to the initial statement, there are of course other ways, apart from the application-oriented one, to walk on through the fascinating landscape of functional analysis and operator theory. In fact, just playing with mathematical ideas may be a highly enjoyable and gratifying activity, in rather the same way as playing a music instrument. Anybody who wants to enjoy just this feeling of mathematics is recommended to return from time to time to the author’s celebrated treatize on nonlinear functional analysis in 5 volumes.

46-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis
47-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operator theory
81-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory