Topological Vector Spaces and Algebras
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230 Lucien Waelbroeck Vrije Universiteit Brussel, BrussellBelgie presently at the Universite Libre de Bruxelles, Bruxelles/Belgique
Topological Vector Spaces and Algebras
Springer-Verlag Berlin· Heidelberg· NewYork 1971
Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich
230 Lucien Waelbroeck Vrije Universiteit Brussel, BrussellBelgie presently at the Universite Libre de Bruxelles, Bruxelles/Belgique
Topological Vector Spaces and Algebras
Springer-Verlag Berlin· Heidelberg· NewYork 1971
AMS Subject Classifications (1970): Primary: 46-02, 46A IS, 46A 99, 46HOS, 46)OS Secondary: 46£40, 46FOS ISBN 3-S40-0S6S0-S Springer-Verlag Berlin . Heidelberg . New York ISBN 0-387-0S6S0-S 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 1971. Library of Congress Catalog Card Number 70-178759. Printed in Germany. Offsetdruck: Julius Beltz, HemsbachiBergstr.
toe h r i s tin e
INTRODUCTION
The lectures associated with these notes were given at the Instituto de Matematica Pura e Aplicada (It1PA) in Rio de Janeiro, during the local winter 1970. To emphasize the properties of topological algebras, the author had started out his lectures with results about topological algebras, and introduced the linear results as he went along. The present exposition is more systematic. It is more or less divided into two parts. The first part contains the first four chapters and could be headed Linear Theory. The second part contains the five remaining chapters and would be headed Algebras. The author has not intended to write a complete survey of the subject. Locally convex spaces have become classical. Nuclear spaces are not used in Part II. Both theories have been omitted. Part I mainly contains structures which are not classical, but which are useful in the development of topological algebra theory according to the author's experience. Other mathematicians will feel that this author has been unfair to their own pet theories .,. A limited number of interesting constructions applicable to topological algebras are developed in Part II. It is hoped that the reader will get an idea of the tools with which one can tackle problems involving topological algebras, also of the problems which can be handled in this way. But the reader must be aware that many interesting results will not be found in these notes. Most chapters are concluded by a section "Notes and Remarks", in which the author speaks about the history, the development of the subject. These Notes are
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