Topological Vector Spaces The Theory Without Convexity Conditions
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639 Norbert Adasch Bruno Ernst DjAfar Keirn
Topological Vector Spaces The Theory Without Convexity Conditions
Springer-Verlag Berlin Heidelberg New York 1978
Authors Norbert Adasch Fachbereich der Mathematik der Universitat Robert Mayer-Str. 10 0-6000 Frankfurt am Main Bruno Ernst Fachbereich Mathematik der Gesamthochschule Warburger Str. 100 0-4790 Paderborn Dieter Keim Fachbereich Mathematik der Universitat Robert Mayer-Str. 10 0-6000 Frankfurt am Main
AMS Subject Classifications (1970): 46-02, 46A05, 46A07, 46A09, 46A15, 46A30 ISBN 3-540-08662-5 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-08662-5 Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin Heidelberg 1978 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
Unserem verehrten Lehrer Gottfried
Kothe
in Dankbarkeit gewidmet
Contents page
Introduction Strings and linear topologies
§
5
§
2
Metrizable topological vector spaces
10
§
3
Projective limits of topological vector spaces
16
§
4
Inductive limits of topological vector spaces
19
§
5
Topological direct sums, strict inductive limits
24
§
6
Barrelled topological vector spaces
31
§
7
The Banach-Steinhaus theorem
38
§
8
Barrelled spaces and the closed graph theorem
41
§
9
Barrelled spaces and the open mapping theorem
47
§ 10
Completeness and the closed graph theorem
51
§ 11
Bornological spaces
60
§ 12
Spaces of continuous linear mappings and their completion
66
§ 13
Quasibarrelled spaces
70
§ 14
Boundedly summing spaces
74
§ 15
Locally topological spaces
79
§ 16
Spaces with an absorbing sequence
84
§ 17
() -locally topological spaces
92
§ 18
(DF)-spaces and spaces with a fundamental sequence of compact sets
98
§ 19
Some examples and counter examples
109
References
118
Subject Index
123
One of the earliest and most important results in functional analysis are the closed graph theorem and the open mapping theorem of Banach and Schauder. Usually these theorems are known for Banach spaces, but already the original version gives their validity for complete metrizable topological vector spaces, a class of spaces including not only the Banach spaces, but also other important spaces for which in general duality methods cannot be applied in the proofs. But this "non convex" approach was dropped in the following development of functional analysis. Starting from Banach spaces the theory turned to the more general class of locally convex spaces, making extensive use of the close interrelation betwee
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