This book is an interesting addition to the literature about locally convex spaces and their applications. On the one hand, it includes a clear, self-contained exposition of the basic theory of topological vector spaces, and on the other hand, it presents many important results that are outside the classical core of the topic. Many examples of functions and sequence spaces are presented in full detail.
There are mainly two novelties in this book. Firstly, in the half of the book, Chapters 4 and 5, are devoted to differential calculus in locally convex spaces and to measure theory on topological vector spaces. The other main novelty is the following: each chapter finishes with a section about complements and exercises. In the complements, the authors include several subsections with more specialized information that surveys, sometimes without proof, relevant recent results related to the content of the chapter. There are two type of exercises, the usual ones that are clearly marked, and exercises that state recent achievements with explicit references to the literature.
The prerequisites of the book are knowledge of mathematical analysis and linear algebra and basic concepts in set topology. Knowledge of functional analysis and measure theory at the university level might be advisable for the reader who really wants to appreciate the whole content. Certain topics, like nuclear Fréchet spaces, bases, or applications to distribution theory or partial differential operators, are not presented in the book. This is understandable, since a selection is necessary. These aspects are treated in other texts. The book has an updated, long list of references that consists of 582 entries. The author index and the subject index are very exhaustive and useful. The book finishes with 6 pages about the history of topological vector spaces, since its beginning in the 1930’s until very recent developments, with precise references.
Chapter 1 contains the fundamentals of the theory of locally convex spaces, including many concrete examples. Convex sets, metrizability, (pre)compact sets, completeness and continuous linear operators are discussed. Several forms of the Hahn-Banach theorem are proved. Separability, fixed point theorems and continuous selections and extensions are presented in the complements, among other topics.
Projective and inductive limits are studied in Chapter 2. Strict inductive limits and inductive limits with compact embeddings are discussed. This chapter also includes some information about tensor products, nuclear spaces, the test space for distributions and absolutely summing operators.
Chapter 3 presents the classical duality theory: different topologies of uniform convergence, the Mackey-Arens theorem, barrelled, bornological and reflexive spaces, weak compactness and the Grothendieck completeness theorem. Other interesting topics are also included: closed graph theorems, compact operators on locally convex spaces and the Fredholm alternative, Baire spaces, Schauder bases, products of copies of the scalar field, etc.
Chapter 4 is devoted to differentiation theory in locally convex spaces. The basics of differential calculus are presented in detail. A key role is played by differentiability with respect to a system of sets (in particular, bounded or compact sets), particular cases of which are Gâteaux, Hadamard and Fréchet differentiability. In the main parts of the chapter, the authors deal with definitions of differentiability that do not require filters and pseudo-topologies. Many examples are included in detail. Composition of differentiable mappings and the inverse function theorem, polynomials and ordinary differential equations in locally convex spaces are discussed in the complements.
The interesting and informative Chapter 5 gives an account of measure theory on locally convex spaces. It requires familiarity with the Lebesgue measure and integration theory. The books by the first author [Gaussian measures. Transl. from the Russian by the author. Providence, RI: American Mathematical Society (1998; Zbl 0913.60035); Measure theory. Vol. I and II. Berlin: Springer (2007; Zbl 1120.28001)] are related to the presented material. The authors discuss cylindrical measures, the Fourier transform, conditions for countable additivity in terms of the Fourier transform, covariance operators, measurable linear functionals and operators, Gaussian measures, convex measures, the central limit theorem, and infinite-dimensional Wiener processes.
This is indeed a good book, well written, that includes much useful material. The basic theory is presented in a clear, understandable way. Moreover, many recent, important, more specialized results are also included with precise references. This book is recommendable for analysts interested in the modern theory of locally convex spaces and its applications, and especially for those mathematicians who might use differentiation theory on infinite-dimensional spaces or measure theory on topological vector spaces.