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

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.

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 |