Approximation and Gâteaux differentiability of convex function in Banach spaces

Shaoqiang Shang and Yunan Cui

Mathematische Nachrichten, 2021, vol. 294, issue 12, 2413-2424

Abstract: This 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 {fn}n=1∞${\big \lbrace f_{n}\big \rbrace} _{n=1}^{\infty }$ of continuous convex functions such that (1) fn(x)≤fn+1(x)≤f(x)$f_{n}(x)\le f_{n+1}(x)\le f(x)$ on X; (2) fn$f_{n}$ is Gâteaux differentiable at all points of a dense open subset of X; (3) fn(x)→f(x)${f_n}(x) \rightarrow f(x)$ on E. Moreover, if X is separable, then there exists a continuous convex function sequence {fn}n=1∞${\big \lbrace f_{n}\big \rbrace} _{n=1}^{\infty }$ such that (1) and (2) are true and fn(x)→f(x)${f_n}(x) \rightarrow f(x)$ on X.

Date: 2021
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