Uniform approximation of convex function in smooth Banach spaces

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, Gateaux differentiable convex function sequences and {g(n)}(n=1)(infinity) such that (1) f(n) < f(n) <= f(n+1) <= f <= g(n+1) <= g(n); (2) f(n)-> f and g(n) -> f uniformly on X*; (3) cl{x* is an element of X* : df(n)(x*) is an element of X} = cl{x* is an element of X* : dg(n)x*) E X} = (C) 2019 Elsevier Inc. All rights reserved.