From Hahn-Banach to Monotonicity
In this new edition of LNM 1693 the essential idea is to reduce questions on monotone multifunctions to questions on convex functions. However, rather than using a “big convexification” of the graph of the multifunction and the “minimax technique”for prov
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Stephen Simons
From Hahn-Banach to Monotonicity
1693 Second Edition
Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris
1693
Stephen Simons
From Hahn-Banach to Monotonicity 2nd, expanded edition
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Stephen Simons Department of Mathematics University of California Santa Barbara, CA 93105-3080 USA [email protected] http://www.math.ucsb.edu/∼simons
1st edition 1998 LNM 1693: Minimax and Monotonicity ISBN 978-1-4020-6918-5
e-ISBN 978-1-4020-6919-2
DOI 10.1007/978-1-4020-6919-2 Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2007942159 Mathematics Subject Classification (2000): 46A22, 49J35, 47N10, 49J52, 47H05 c 2008 Springer Science + Business Media B.V., 1998 Springer-Verlag Berlin Heidelberg No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Cover design: WMXDesign GmbH Printed on acid-free paper 987654321 springer.com
For Jacqueline whose support and patience are unbounded.
Preface
A more accurate title for these notes would be: “The Hahn–Banach–Lagrange theorem, Convex analysis, Symmetrically self–dual spaces, Fitzpatrick functions and monotone multifunctions”. The Hahn–Banach–Lagrange theorem is a version of the Hahn–Banach theorem that is admirably suited to applications to the theory of monotone multifunctions, but it turns out that it also leads to extremely short proofs of the standard existence theorems of functional analysis, a minimax theorem, a Lagrange multiplier theorem for constrained convex optimization problems, and the Fenchel duality theorem of convex analysis. Another feature of the Hahn–Banach–Lagrange theorem is that it can be used to transform problems on the existence of continuous linear functionals into problems on the existence of a single real constant, and then obtain a sharp lower bound on the norm of the linear functional satisfying the required condition. This is the case with both the Lagrange multiplier theorem and the Fenchel duality theorem applications mentioned above. A multifunction from a Banach space into the subsets of its dual can, of course, be identified with a subset of the product of the space with its dual. Simon Fitzpatrick defined a convex function on this product corresponding with any such multifunction. So part of these notes is devoted to the rather special convex analysis for the product of a Banach space with its dual. The product of a Banach space with its dual is a special case of a “symmetrically self–dual space”. The advantage of going to this slightly higher level of abstraction is not only that it leads to more general results but, more to the point, it
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