Functors and Categories of Banach Spaces Tensor Products, Operator I
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651 Peter W. Michor
Functors and Categories of Banach Spaces Tensor Products, Operator Ideals and Functors on Categories of Banach Spaces
SpringerVeriag Berlin Heidelberg New York 1978
Author Peter W. Michor Institut fur Mathematik Universitat Wien Strudlhofgasse 4 A-1090 Wien
Library of Congress Cataloging in Publication Data
Michor, Peter W 1949Funct.or-s and categories ot Banach spaces. (Lecture notee in mathematics ; 651) Bibliography: p , Includes index. 1. Banach spaces. 2. Categories (Mathematics) 3. Functor theory. I. Title. II. Series: Lecture notes in mathematics (Berlin) ; 651. QA3.L28 no. 651 (QA322.2J 510' .8s (515'. 73J 78-6814 ISBN 0-387-08764-8
AMS Subject Classifications (1970): 46M05, 46M15 ISBN 3-540-08764-8 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-08764-8 Springer-Verlag New York Heidelberg Bertin 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
Preface
1
The aim of this book is to develop the theory of Banach operator ideals and metric tensor products along categorical lines: these two classes of mathematical objects are endofunctors on the category Ban of all Banach spaces in a natural way and may easily be characterized among them (§4). Up to now they were investigated with methods of functional analysis in a sort of ad hoc manner and with an outlook to special properties; here they are subject to several categorical and universal constructions: Kan extensions from the subcategory of finite dimensional spaces are studied in §2 and applied to tensor products and operator ideals in §§ 4,5,6 and give rise to the reappearance of the &norms in the sense of Grothendieck and to minimal and maximal operator ideals in the sense of Pietsch.
Duality for co and contravariant functors is studied in §3 (and some new and deep results are derived on it) and is applied to tensor products and operator ideals in §§ 4,5,6: duality is the link between the two notions. Several other constructions of sub and quotient functors induced by canonical adjoint relations are used to (co) reflect all appearing functors back to tensor products and operator ideals (§§4,5,6). In §7 we introduce (as an example) a new class of tensor prOducts, the projective (p,r,s)tensor product, which is a link between the (p,r,s)absolutely summing, nuclear and integral operator ideals and we use it to derive a lot of new relations between these operator ideals from existing ones. The whole subject although sometimes technical and complicated seem
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