Approximation and Gâteaux differentiability of convex function in Banach spaces

Abstract

AbstractThis paper shows that if f is a convex continuous function on a Gâteaux differentiability space X, then for any separable closed subspace E of X, there exists a sequence of continuous convex functions such that (1) on X; (2) is Gâteaux differentiable at all points of a dense open subset of X; (3) on E. Moreover, if X is separable, then there exists a continuous convex function sequence such that (1) and (2) are true and on X.