Additive Subgroups of Topological Vector Spaces
The Pontryagin-van Kampen duality theorem and the Bochner theorem on positive-definite functions are known to be true for certain abelian topological groups that are not locally compact. The book sets out to present in a systematic way the existing materi
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Wojciech Banaszczyk
Additive Subgroups of Topological Vector Spaces
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Author Wojciech Banaszczyk Institute of Mathematics Lodz University Banacha 22 90-238 Lodz, Poland
Mathematics Subject Classification (1980): llH06, 22-02, 22AIO, 22A25, 22B05, 40105,43-02, 43A35, 43A40, 43A65, 46A12, 46A25, 46B20, 47BIO, 52A43, 60Bl5
ISBN 3-540-53917-4 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-53917-4 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions ofthe German Copyright Law of September 9, 1965, in its current version, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1991 Printed in Germany Printing and binding: Druckhaus Beltz, HemsbachlBergstr. 2146/3140-543210 - Printed on acid-free paper
PREFACE
In the commutative harmonic analysis there are at least two important theorems that make sense without the assumption of the local compactness of the group and the existence of the Haar measure: the Pontryagin-van Kampen duality theorem and the Bochner theorem on positive-definite functions. The Pontryagin-van Kampen theorem is known to be true e.g. for Banach spaces, products of locally compact groups or additive subgroups and quotients of nuclear Frechet spaces. The Bochner theorem remains valid for locally convex spaces over p-adic fields, for nuclear locally convex spaces (the Minlos theorem), their subgroups and quotients. These lecture notes are an attempt of clearing up the existing material and of determining the "natural" limits of the applicability of the theory. Pontryagin duality is discussed in chapter 5 and the Bochner theorem in chapter 4. Our exposition is based on the notion of a nuclear group. Roughly speaking, nuclear groups form the smallest class of abelian topological groups which contains locally compact groups and nuclear spaces and is closed with respect to the operations of taking subgroups, Hausdorff quotients and arbitrary products. The definition and basic properties of nuclear groups are gathered in chapter 3. It turns out that, from the point of view of continuous characters, nuclear groups inherit many properties of locally compact groups. In chapter 2 we show that the assumption of nucleari ty is essential: if a separable Frechet space E is not nuclear, it contains a discrete additive subgroup K such that the quotient group ElK does not admit any non-trivial continuous unitary representations. In section 10 we apply nuclear groups to obtain answer to an old problem of S. Ulam on rearrangement of series in topological grou
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