Convex Functions, Monotone Operators and Differentiability
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1364
Robert R. Phelps
Convex Functions, Monotone Operators and Differentiability 2nd Edition
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Author Robert R. Phelps Department of Mathematics GN-50 University of Washington Seattle, WA 98195-0001, USA
Cover Graphic by Diane McIntyre
Mathematics Subject Classification (1991): 46B20, 46B22, 47H05, 49A29, 49A51 , 52A07 ISBN 3-540-56715-1 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-56715-1 Springer-Verlag New York Berlin Heidelberg Library of Congress Cataloging-in-Publication Data Phelps, Robert R. (Robert Ralph), 1926 Convex functions, monotone operators, and differentiability/Robert R. Phelps. 2nd. ed. p. cm. - (Lecture notes in mathematics; 1364) Includes bibliographical references and index. ISBN 0-387-56715-1 1. convex functions. 2. Monotone operators. 3. Differentiable Functions. I. Title. II. Series: Lecture notes in mathematics (Springer Verlag); 1364. QA3.L28 no. 1364 (QA331.5) 510 s-dc20 (515'.8) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1989, 1993 Printed in Germany Printing and binding: Druckhaus Beltz, HemsbachlBergstr. 2146/3140-543210 - Printed on acid-free paper
PREFACE In the three and a half years since the first edition to these notes was written there has been progress on a number of relevant topics. D. Preiss answered in the affirmative the decades old question of whether a Banach space with an equivalent Gateaux differentiable norm is a weak Asplund space, while R. Haydon constructed some very ingenious examples which show, among other things, that the converse to Preiss' theorem is false. S. Simons produced a startlingly simple proof of Rockafellar's maximal monotonicity theorem for subdifferentials of convex functions. G. Godefroy, R. Deville and V. Zizler proved an exciting new version of the Borwein-Preiss smooth variational principle. Other new contributions to the area have come from J. Borwein, S. Fitzpatrick, P. Kenderov, 1. Namioka, N. Ribarska, A. and M. E. Verona and the author. Some of the new material and substantial portions of the first edition were used in a one-quarter graduate course at the University of Washington in 1991 (leading to a number of corrections and improvements) and some of the new theorems were presented in the Rainwater Seminar. An obvious improvement is due to the fact that I learned to use '!EX. The task of converting the original MacWrite text to '!EXwas performed by Ms. M
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