Convex Functions, Monotone Operators and Differentiability
These notes start with an introduction to the differentiability of convex functions on Banach spaces, leading to the study of Asplund spaces and their intriguing relationship to monotone operators (and more general set-values maps) and Banach spaces with
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1364 Robert R. Phelps
Convex Functions, Monotone Operators and Differentiability
Springer-Verlag Berlin Heidelberg GmbH
Author
Robert R. Phelps Department of Mathematics GN-50, University of Washington Seattle, WA 98195, USA
Mathematics Subject Classification (1980): 46B20, 46B22, 47H05, 49A29, 49A51,52A07
ISBN 978-3-540-50735-2 DOI 10.1007/978-3-662-21569-2
ISBN 978-3-662-21569-2 (eBook)
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© Springer-Verlag Berlin Heidelberg 1989 Originally published by Springer-Verlag Berlin Heidelberg New York in 1989 2146/3140-543210
PREFACE These notes had their genesis in a widely distributed but unpublished set of notes Differentiability of convex functions on Banach spaces which I wrote in 1977-78 for a graduate course at University College London (UCL). Those notes were largely incorporated into J. Giles' 1982 Pitman Lecture Notes Convex analysis with application to differentiation of convex functions. In the course of doing so, he reorganized the material somewhat and took advantage of any simpler proofs available at that time. I have not hesitated to return the compliment by using a few of those improvements. At my invitation, R. Bourgin has also incorporated material from the UeL notes in his extremely comprehensive 1983 Springer Lecture Notes Geometric aspects of convex sets with the Radon-Nikodym property. The present notes do not overlap too greatly with theirs, partly because of a substantially changed emphasis and partly because I am able to use results or proofs that have come to light since 1983. Except for some subsequent reVISIOns and modest additions, this material was covered in a graduate course at the University of Washington in Winter Quarter of 1988. The students in my class all had a good background in functional analysis, but there is not a great deal needed to read these notes, since they are largely self-contained; in particular, no background in convex functions is required. The main tool is the separation theorem (a.k.a. the Hahn-Banach theorem); like the standard advice given in mountaineering classes (concerning the all-important bowline for tying oneself into the end of the climbing rope), you should be able to employ it using only one hand while standing blindfolded in a cold shower. These notes have been influenced very considerably by frequent conversations with Isaac Namioka (who has an almost notorious instinct for simplifying proofs) as well as occasional conversations with Terry RockafelIar; I am grateful to them both.
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