Convex Functions, Monotone Operators and Differentiability

Robert R. Phelps (auth.)9780387507354, 0387507353

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 the Radon-Nikodym property. While much of this is classical, some of it is presented using streamlined proofs which were not available until recently. Considerable attention is paid to contemporary results on variational principles and perturbed optimization in Banach spaces, exhibiting their close connections with Asplund spaces. An introductory course in functional analysis is adequate background for reading these notes which can serve as the basis for a seminar of a one-term graduate course. There are numerous excercises, many of which form an integral part of the exposition.

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Table of contents :
Front Matter….Pages I-IX
Convex functions on real Banach spaces….Pages 1-16
Monotone operators, subdifferentials and Asplund spaces….Pages 17-39
Lower semicontinuous convex functions….Pages 40-63
A smooth variational principle and more about Asplund spaces….Pages 64-71
Asplund spaces, the Radon-Nikodym property and optimization….Pages 72-89
Gateaux differentiability spaces….Pages 90-96
A generalization of monotone operators: Usco maps….Pages 97-103
Notes and Remarks….Pages 104-107
Back Matter….Pages 108-118