Topological vector spaces. 2nd ed. (English) Zbl 0983.46002

Schaefer, Helmut H.; Wolff, M. P.

Graduate Texts in Mathematics. 3. New York, NY: Springer. vi, 346 p. (1999).

The reliable textbook, highly estimated by several generations of students since its first edition in 1966 (cf. Zbl 0141.30503) is appeared now in extended, second edition. This book is a systematic text on topological vector spaces, including locally convex spaces, ordered topological vector spaces, and topological algebras, but the reader should be familiar in advance with the elementary facts on Banach and Hilbert spaces, to understand the general functional analytical notions and constructions presented here. The second edition was extended by a chapter on \(C^*\)- and \(W^*\)-algebras. The book will not and cannot replace other books on Banach spaces, operator spaces or on special function spaces, as the \(L_p\)-spaces, and, as the title says, the isometric theory of Banach spaces is not touched. The book is organized as follows:
Chapter 1: Topological vector spaces,
Chapter 2: Locally convex spaces,
Chapter 3: Linear mappings,
Chapter 4: Duality,
Chapter 5: Order structures,
Chapter 6: \(C^*\)- and \(W^*\)-algebras,
Appendix: Spectral properties of positive operators.
The book contains a large number of interesting exercises. Besides of other good books on linear topological spaces the book of Schaefer and Wolff keepts worth reading.

Reviewer: Heinz Junek (Potsdam)

MSC:

46A03	General theory of locally convex spaces
46-02	Research exposition (monographs, survey articles) pertaining to functional analysis
46A04	Locally convex Fréchet spaces and (DF)-spaces
46A11	Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.)
46A25	Reflexivity and semi-reflexivity
46L05	General theory of \(C^*\)-algebras
46A40	Ordered topological linear spaces, vector lattices
46A20	Duality theory for topological vector spaces
46A30	Open mapping and closed graph theorems; completeness (including \(B\)-, \(B_r\)-completeness)
46A32	Spaces of linear operators; topological tensor products; approximation properties
46B42	Banach lattices
46A55	Convex sets in topological linear spaces; Choquet theory
47B60	Linear operators on ordered spaces
46F05	Topological linear spaces of test functions, distributions and ultradistributions

Keywords:

locally convex spaces; projective topology; injective topology; barrelled spaces; bornological spaces; Banach’s homomorphism theorem; Banach-Steinhaus theorem; tensor products; nuclear mappings and spaces; duality; Mackey-Arens theorem; reflexive spaces; theorems of Grothendieck; Banach-Dieudonné; Krein-Smulian; open mapping and closed graph theorems; absolute summability; weak compactness; Eberlein-Krein theorem; ordered topological vector spaces; positive operators; theorems of Stone-Weierstrass and Kakutani; \(C^*\)- and \(W^*\)-algebras; Gelfand’s theorem; Gelfand-Naimark-Segal theorem; Jordan decomposition; trace class operators; factors; spectral properties of positive operators

Citations:

Zbl 0212.14001; Zbl 0217.16002; Zbl 0435.46003; Zbl 0141.30503