Topological Vector Spaces I
It is the author's aim to give a systematic account of the most im portant ideas, methods and results of the theory of topological vector spaces. After a rapid development during the last 15 years, this theory has now achieved a form which makes such an
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Editors
M. Artin s. S. Chern J. L. Doob A. Grothendieck E. Heinz F. Hirzebruch L. Hormander S. Mac Lane W. Magnus C. C. Moore J. K Moser M. Nagata W. Schmidt D. S. Scott 1. Tits B. L. van der Waerden Managing Editors
M. Berger
B. Eckmann
S. R. S. Varadhan
Gottfried Kothe
Topological Vector Spaces I
Translated by D. J. H. Garling Second printing, revised
Springer-Verlag Berlin· Heidelberg· New York 1983
Prof. Dr. Dr. h. c. Gottfried Kothe Institut fUr angewandte MathematIk
der Johann-Wolfgang-Goethe-Universitat, Frankfurt am Main
Translation of Topologische Lineare Riiume I, 1966 (Grundlehren der mathematischen Wissenschaften, Vol. 107)
ISBN-13: 978-3-642-64990-5 DOT: 10.1007/978-3-642-64988-2
e-ISBN-13: 978-3-642-64988-2
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© Springer-Verlag Berlin, Heidelberg 1983 Softcover reprint of the hardcover 1st edition 1983 The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Offsetprinting and Bookbinding: Zechnersche Buchdruckerei Speyer 2141/3020-543210
Preface to the First Edition It is the author's aim to give a systematic account of the most important ideas, methods and results of the theory of topological vector spaces. After a rapid development during the last 15 years, this theory has now achieved a form which makes such an account seem both possible and desirable. This present first volume begins with the fundamental ideas of general topology. These are of crucial importance for the theory that follows, and so it seems necessary to give a concise account, giving complete proofs. This also has the advantage that the only preliminary knowledge required for reading this book is of classical analysis and set theory. In the second chapter, infinite dimensional linear algebra is considered in comparative detail. As a result, the concept of dual pair and linear topologies on vector spaces over arbitrary fields are introduced in a natural way. It appears to the author to be of interest to follow the theory of these linearly topologised spaces quite far, since this theory can be developed in a way which closely resembles the theory of locally convex spaces. It should however be stressed that this part of chapter two is not needed for the comprehension of the later chapters. Chapter three is concerned with real and complex topological vector spaces. The classical results of Banach's theory are given here, as are fundamental results about convex sets in infinite dimensional spaces. The subsequent chap
