Geometry of Banach Spaces-Selected Topics
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485 III
Im
Joseph Diestel
Geometry of Banach Spaces Selected Topics
Springer-Verlag 9 York 1975 Berlin. Heidelberg New
Author Prof. Joseph Diestel Department of Mathematics Kent State University Kent, Ohio 44242 USA
Library of Congress Cataloging in Publication Data
Diestel, Joseph, 19~BGeometry of Banach spaces. (Lecture notes in mathematics ; 485) Bibliography: p. Includes index. 1. Banach spaces. 2. Vector-valued measures. I. Title. II. Series: Lecture notes in mathematics
(Berlin) ; /+85.
QA3.L28 no. 485
EQA322.2]
510'.8s c515w.73~
75-26821
AMS Subject Classifications (1970): 28A45, 46 B10, 46 B99 ISBN 3-540-07402-3 Springer-Verlag Berlin Heidelberg 9 New 9 York ISBN 0-387-0?402-3 Springer-Verlag New York Heidelberg 9 Berlin 9 This work is subject to copyright. All rights are reserved, wh~ther 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 w 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. 9 by Springer-Verlag Berlin 9Heidelberg1975 Printed in Germany Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
Preface
These notes were the subject of lectures given at Kent State University during the 1973-74 academic year. the geometry of Banach spaces
At that time, it was already clear that
(in the form of convexity and smoothness
type
considerations)
would play a central role in the theory of Radon-Nikodym
differentiation
for vector-valued
course:
to acquaint my students
the geometry of Banach spaces.
measures.
This was the object of the
(and, to a large extent, myself) with Naturally,
the logical
courses was a discussion of the Radon-Nikodgm
finish to the
theorem viewed
from a
pure%y geometric perspective. Some words about the organization
of the notes.
The first chapter deals with the plenitude of support functionals closed bounded convex subsets of a Banaeh space. the Bishop-Phelps weak compactness. modern functional
subreflexivity
Two results are focal:
theorem and James' characterization
of
I feel that these are among the deepest results of analysis and have tried throughout
them whenever possible.
When the Deity allowed
be proved, He meant for them to be usedl application
to operators attaining
topological
tensor products
concerning prerequisites a one-time affair.
to
the notes to apply
for theorems like these to
This chapter is closed with an
their norm which uses the theory to
for its proof;
this is the only excursion
outside of elementary
functional
analysis and is
The principle purpose here is to highlight
restriction placed upon a Banach space
the severe
(or pair of Banach spaces)
operator attain its norm; it also is an interesting
that every
application of James'
theorem. Chapter Two deals with the basics of convexity and smoothness. prov
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