Interpolation Functors and Duality
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1208 Sten Kaijser Joan Wick Pelletier
Interpolation Functors and Duality
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Authors
Sten Kaijser Uppsala University, Department of Mathematics Thunbergsviigen 3, S-752 38 Uppsala, Sweden Joan Wick Pelletier York University, Department of Mathematics 4700 Keele Street, North York, Ontario, Canada, M3J 1P3
Mathematics Subject Classification (1980): 46M 15, 46M35 ISBN 3-540-16790-0 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-16790-0 Springer-Verlag New York Berlin Heidelberg
Library of Congress Cataloging-in-Publication Data. Kaijser, Sten.lnterpolation functors and duality. (Lecture notes in mathematics; 1208) Bibliography: p.lncludes index. 1. Lineartopological spaces. 2. Functor theory. I. Pelletier, Joan Wick, 1942-.11. Title. III. Series: Lecture notes in mathematics (Springer-Verlag); 1208. QA3.L28 no. 1208510 s 86-20242 [QA322] [515.7'3] ISBN 0-387-16790-0 (U.S.) This work is subject to copyriqht, 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 "Verwertungsgesellschaft Wort", Munich.
© Springer-Verlag Berlin Heidelberg 1986 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
CONTENTS
O.
Introduction
1
PART I I.
Preliminaries 1. 2. 3.
11.
7
12 16
The Real Method 1. 2. 3.
III.
The Setting Doolittle Diagrams, Couples, and Regular Couples Interpolation Spaces
The J- and K-methods The Duality Theorem The Equivalence Theorem
18 22 25
The Complex Method 1. 2.
The General Duality Theorem The Duality Theorem
33 38
PART II IV.
Categorical Notions 1. 2. 3. 4. 5.
V.
44
50
54 58 64
Finite Dimensional Doolittle Diagrams 1. 2. 3. 4.
VI.
Categories of Doolittle Diagrams Doolittle Diagrams of Banach Spaces Limits, Colimits, and Morphisms Functors and Natural Transformations Interpolation Spaces and Functors
I-dimensional Doolittle Diagrams and Applications The Structure Theorem Operators of Finite Rank Applications
73 79 84 86
Kan Extensions 1. 2. 3. 4. 5.
Definition Examples Computable Functors Aronszajn-Gagliardo Functors Computability of LanA
93 94 99 100 104
IV
VII.
Duality 1. 2. 3. 4. 5.
Dual Functors Descriptions of the Dual Functors Duality for Computable Functors Approximate Reflexivity Duals of Interpolation Functors
106 108 111 115 117
PART III VIII.
More About Duality 1. 2.
IX.
Comparison of Parts I and II Quasi-injectivity and Quasi-projectivity
123 126
The Classical Methods from a Categorical Viewpoint 1. 2. 3. 4.
Review of Results The Real Method Revisited The Complex Method Revisited The Dual Functor of C
e
132 133 143 154
Bibliography
160
List of Special Symbols and Abbreviations
162
Index
165
CHAPTER 0
INTRODUCTION
Duality is one of th
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