This is the second edition of the book published by the same authors with the same title in 1985 [L. Narici and E. Beckenstein, “Topological vector spaces” (Pure and applied mathematics 95; New York-Basel: Marcel Dekker, Inc.) (1985; Zbl 0569.46001)]. This is a nicely written, easy to read expository book of the classical theory of topological vector spaces. As explicitly mentioned by the authors, the book is addressed to beginners, who are only familiar with set topology and linear algebra. The proofs are complete and very detailed. The authors mainly concentrate on locally convex spaces and present the material around four basic principles of functional analysis, namely, the Hahn-Banach theorem, the Banach-Steinhaus theorem, the Krein-Milman theorem and the closed graph theorem. Each chapter finishes with a large number of exercises that are distributed in two categories of different level of difficulty. Hints are given for many exercises. The comprehensive exposition and the quantity and variety of exercises makes the book really useful for beginners, and makes the material more easily accessible than the excellent classical monographs by Köthe or Schaefer.
The new features of this edition, on top of a partial update of the reference list, rearrangement and rewriting, are the following: Chapter 7 on the Hahn-Banach theorem has been expanded and new, interesting material has been added; in particular, the connection with the axiom of choice is explained in section 7.11. Especially informative for the student are section 7.9 about the origin of the theorem and section 7.13 about the life and work of Eduard Helly. There is also a new Chapter 10 about vector valued Hahn-Banach extension theorems. Section 11.1, at the beginning of the chapter on uniform boundedness principles, is also of historical character and is dedicated to the Scottish Café in Lwów, where Banach and his group met in the 1930’s. Finally, an enlarged presentation of Banach-Stone theorems that includes discussions of separating maps, vector valued versions and non-archimedean theorems, is presented in Chapter 9. The authors have made relevant contributions on these topics.
Here is a brief description of the content of the 16 chapters of the book. Chapter 1 presents background on topology, algebra and measure theory. Chapters 2 and 3 are dedicated to commutative topological groups, including metrizability, completeness and (pre)compactness. The elementary theory of locally convex topological vector spaces, including seminorms, bounded sets and some examples of spaces of continuous functions, is presented in Chapter 4, 5 and 6. We have already explained the main points of the interesting Chapter 7 on the Hahn-Banach theorem. Duality, dual pairs, topologies of the dual pairs and transposed maps are the content of Chapter 8. The first part of Chapter 9 discusses the Krein-Milman theorem and the second part the Banach-Stone theorem. Different classes of locally convex spaces, like barrelled spaces (and the Banach-Steinhaus theorem), bornological spaces, inductive limits and reflexive spaces are studied in Chapter 11, 12, 13 and 16. The closed graph theorem of de Wilde on webbed spaces is presented in Chapter 14. Other closed graph theorems are mentioned. The last chapter 16 is dedicated to strict and uniform convexity on Banach spaces and best approximation. Several important topics like Schwartz spaces, nuclear spaces, tensor products, approximation properties, or applications to distribution theory are not mentioned. The reader interested in the modern theory of locally convex spaces, including Fréchet and (DF)-spaces, nuclear spaces, Köthe echelon spaces, the isomorphic classification of subspaces and quotients of power series spaces, the splitting of short exact sequences and the applications to complex analysis, Schwartz distribution theory and linear partial differential operator, should look carefully at part IV of the book [R.Meise and D.Vogt, Introduction to Functional Analysis. Oxford Graduate Texts in Mathematics. 2. Oxford: Clarendon Press (1997; Zbl 0924.46002)].
Summarizing, this is a well-written book, with comprehensive proofs, many exercises and informative new sections of historical character, that presents in an accessible way the classical theory of locally convex topological vector spaces and that can be useful especially for beginners interested in this topic.