Geometry and Probability in Banach Spaces
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852 Lau rent Schwartz
Geometry and Probability in Banach Spaces Notes by Paul R. Chernoff
Springer-Verlag Berlin Heidelberg NewYork 1981
Author Laurent Schwartz Centre de Mathematiquea de l'Ecole Polytechnique 91128 Palaiseau Cedex, France Paul R. Chernoff Dept. of Mathematics, University of California Berkeley, CA 94720, USA
AMS Subject Classifications (1980): 46B20, 47BlO, 60B11 ISBN 3-540-10691-X Springer-Verlag Berlin Heidelberg New York ISBN 0-387-10691-)( Springer-Verlag New York Heidelberg Berlin 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 1981 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
PREFACE
These Notes correspond to a course of lectures I gave at the University of California, Berkeley, in April-May 1978.
I tried to
present, in these lectures, the main results of geometry and probability in Banach spaces, which have been the material of several years of the Seminaire de 1'Ecole Polytechnique. task!
A lot of material in ashortctime!
Difficult
It was possible to state
a great number of theorems, and to prove a large part of them. Of course, the longest proofs have been omitted.
However, I believe
that somebody who seriously attended the lectures or who reads these Notes will be able to work by himself in this theory. want to say that I was delighted by the atmosphere in the audience; people seemed to enjoy the lectures very much, and surely I enjoyed myself!
Paul CHERNOFF gives here a very good account of the series
of lectures, with a nice expression of his personal taste; I want to thank him very much!
I N T ROD U C T ION
As said in the Preface, these Lectures summarize a great number of results given in several years of seminaires of the Ecole Poly technique, Palaiseau, France. They cover relationships between geometrical properties, properties of functional analysis, probabilistic properties in Banach spaces, frequently appearing a priori as completely independent of each other. After the brilliant past of the Banach spaces with the Polish
specially Banach,
these spaces were a little abandonned for general locally convex topological vector spaces after World War II (in particular with applications to distributions, and Grothendieck's results about nuclear spaces). But Banach spaces came again, for more refined results, in the 60's, with Lindenstrauss, zynski and others, turning around the LP spaces ; the present subject goes into this direction. Many of the results given here have been found by mathematicians of the French school, in particular Bernard Maurey and Gilles Pisier. Chapter I.
Lecture 1 gives a r
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