Geometry and Nonlinear Analysis in Banach Spaces
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1131 Kondagunta Sundaresan Srinivasa Swaminathan
Geometry and Nonlinear Analysis in Banach Spaces
Spri nger-Verlag
Berlin Heidelberg New York Tokyo
Authors
Kondagunta Sundaresan Department of Mathematics, Cleveland State University Cleveland, Ohio 44115, USA Srinivasa Swami nathan Department of Mathematics, Statistics and Computing Science, Dalhousie University Halifax, Nova Scotia B3H 4H8, Canada
Mathematics Subject Classification (1980): primary: 46B20, 58B10, 58C25 secondary: 41 A65 ISBN 3-540-15237-7 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-15237-7 Springer-Verlag New York Heidelberg Berlin Tokyo 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 "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1985 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
CONTENTS Introduction 1.
1.3
Some Geometric Properties of Banach Spaces Finite Representation of a Banach Space Multilinear Forms and Differential Concepts in Banach Spaces
3 4 8
Smoothness Classification of Banach Spaces 2.1 2.2 2.3 2.4
3.
1
Basic Definitions and Geometric Properties 1.1 1.2
2.
Page
Differentiability Differentiability UFk -Smooth Spaces Classification of
Properties of Norms of Norms in Classical Banach Spaces and Ultrapowers Superreflexive Spaces
12 20 27 35
Smooth Partitions of Unity on Banach Spaces
3.1 3.2 3.3
3.4
4.
S-categories, Smooth Pairs and S-partitions of Unity A Nonlinear Characterization of Superreflexive Spaces Functions on Banach Spaces with Lipschitz Derivatives Miscellaneous Applications
37
47 56
61
Smoothness and Approximation in Banach Spaces
4.1
4.2
4.3
4.4 4.5 Appendix A.O A.l A.2 A.3
Polynomial Algebras on a Banach Space Approximation by Smooth Functions Extensions of Bernstein's Theorem to Infinite Dimensional Banach Spaces Analytic Approximation in Banach Spaces Simultaneous Approximation of Smooth Mappings
67 70 76
86 94
Infinite Dimensional Differentiable Manifolds Introduction Pre liminaries Differentiability in a Half-space Differentiable Manifolds Modelled on Banach Spaces
97 97
99 100
References
110
Index
114
List of Symbols
116
INTRODUCTION This monograph is based in part on a series of lectures given by the first author at Cleveland State University and at Texas A & M University since 1980, and in part on the lectures given by the second author at Dalhousie University and the Australian National University since 1981.
It concerns the development of certain topics in differ-
ential nonlinear analysis in infinite dimensional real Banach spaces. Our motivation derives from the rich and elegant theory of nonlinear analysis in finite dimensional
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