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Zbl 0983.46002
Schaefer, Helmut H.; Wolff, M.P.
Topological vector spaces. 2nd ed. (English)
[B] Graduate Texts in Mathematics. 3. New York, NY: Springer. vi, 346 p. DM 119.00; öS 869.00; sFr. 108.00; \sterling 46.00; \$ 59.95 (1999). ISBN 0-387-98726-6/hbk

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:\par Chapter 1: Topological vector spaces,\par Chapter 2: Locally convex spaces,\par Chapter 3: Linear mappings,\par Chapter 4: Duality,\par Chapter 5: Order structures,\par Chapter 6: $C^*$- and $W^*$-algebras,\par Appendix: Spectral properties of positive operators.\par 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.
[Heinz Junek (Potsdam)]

MSC 2000:: *46A03 General theory of locally convex spaces
46-02 Research monographs (functional analysis)
46A04 Locally convex Frechet spaces, etc.
46A11 Spaces determined by compactness or summability properties
46A25 Reflexivity of topological linear spaces
46L05 General theory of C*-algebras
46A40 Ordered topological linear spaces
46A20 Duality theory of topological linear spaces
46A30 Open mappings and closed graph theorems, etc.
46A32 Topological linear spaces of linear operators, etc.
46B42 Banach lattices
46A55 Convexity in topological linear spaces
47B60 Operators on ordered spaces
46F05 Topological linear spaces of test functions and distributions

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

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