Uniform approximation of convex function in smooth Banach spaces

Abstract

This article is devoted to smooth approximation of convex functions on Banach spaces with smooth norm. We prove that if X⁎ is a smooth space and f is a w⁎-lower semicontinuous Lipschitzian convex function on X⁎, then there exist two w⁎-lower semicontinuous, Gâteaux differentiable convex function sequences {fn}n=1∞ and {gn}n=1∞ such that (1) fn≤fn+1≤f and f≤gn+1≤gn; (2) fn→f and gn→f uniformly on X⁎; (3) cl{x⁎∈X⁎:dfn(x⁎)∈X}=cl{x⁎∈X⁎:dgn(x⁎)∈X}=X⁎.