Convex Functions, Monotone Operators and Differentiability

✍ Scribed by Robert R. Phelps

• Year: 1989
• Tongue: English
• Leaves: 123
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Contents......Page 4
1. Convex functions on real Banach spaces......Page 9
2. Monotone operators, subdifferentials and Asplund spaces......Page 25
3. Lower semicontinuous convex functions......Page 48
4. A smooth variational principle and more about Asplund spaces......Page 72
5. Asplund spaces, the Radon-Nikodym property and optimization......Page 80
6. Gateaux differentiability spaces.......Page 98
7. A generalization of monotone operators: Usco maps......Page 105
8. Notes and Remarks......Page 112
References......Page 116
Index......Page 121
Index of Symbols......Page 123