Differentiability And Approximation Of Convex Functions In Banach Spaces

This dissertation mainly focus on equi-continuity of convex functions, characterization of differentiability of convex functions, smooth approximation of extended real-valued proper convex functions in Asplund spaces and lower semi-continuous functions approximated by A-convex functions.This dissertation includes seven chapters altogether.Chapter I. Introduction presents a survey of the study of differentiability and smoothness approximation of convex functions and A-convex functions in infinite dimensional spaces, outline of this dissertation and some related notions and notations which pass through this dissertation.Chapter II. Equi-continuity of Convex Functions and Its Applications generalizes Banach-Steinhaus theorem from linear functionals to convex functions, and presents some characterizations, simple properties and an application of equi-continuity of convex functions.Chapte III. Approximation and Differentiability of convex functions presents a characterization of differentiability of convex functions using approximation of convex function, and some characterizaitons of Asplund spaces by equi-continuity of convex functions.Chapter IV. Smoothness Approximation of Convex Functions in Asplund Spaces shows that for each extended real-valued proper l.s.c. convex function / on an Aslund space E, there exists a sequence of nonde-creasing, Lipschitzian convex functions {fn}, each of which is Frechet differ-entiable on some dense open subset of E such that fn → f, pointwisely. If / is b-Lipschitzian, fn→f uniformly on every bounded subset of E, and alse shows its dual version.Chapter V. Approximation of Convex Functions in Asplund Generated Spaces shows that the main result of chapter IV is valid in As-plund generated spaces.Chapter VI. Smoothnees Approximation of Lower Semi-Continuous Functions presents that for each l.s.c. function / on Banach space E which admits a LUR and smoothness equivalent norm, there exists a sequence of A-convex functions {fn},which are superdifferentiable on E and bounded on every bounded subset of E, such that , pointwisely. When E is a LUR and Asplund space, by the main result of chapter IV, we obtain such functions {fn} which is Frechet differentiable on some dense open subset of E for each n.Chapter VII. Notes and Remarks presents the main results of chapter III are valid in the sense of differentiability, and show a counterexample that give a negative answer to a clue which wants to show the sufficiency of the main result of chapter IV is ture.