Gateaux Differentiability of Convex Functions and Weak Dentable Set in Nonseparable Banach Spaces

Shaoqiang Shang,Yunan Cui

doi:10.1155/2019/6852859

Abstract

In this paper, we prove that if C⁎⁎ is a ε-separable bounded subset of X⁎⁎, then every convex function g≤σC is Ga^teaux differentiable at a dense Gδ subset G of X⁎ if and only if every subset of ∂σC(0)∩X is weakly dentable. Moreover, we also prove that if C is a closed convex set, then dσC(x⁎)=x if and only if x is a weakly exposed point of C exposed by x⁎. Finally, we prove that X is an Asplund space if and only if, for every bounded closed convex set C⁎ of X⁎, there exists a dense subset G of X⁎⁎ such that σC⁎ is Ga^teaux differentiable on G and dσC⁎(G)⊂C⁎. We also prove that X is an Asplund space if and only if, for every w⁎-lower semicontinuous convex function f, there exists a dense subset G of X⁎⁎ such that f is Ga^teaux differentiable on G and df(G)⊂X⁎.

Highlights

Introduction and PreliminariesLet (X, ‖⋅‖) denote a real Banach space
We prove that X is an Asplund space if and only if, for every w∗-lower semicontinuous convex function f, there exists a dense subset G of X∗∗ such that f is Ĝateaux differentiable on G and df(G) ⊂ X∗
Let N, R, and R+ denote the sets of natural number, reals, and nonnegative reals, respectively

Summary

Introduction and Preliminaries

Let (X, ‖⋅‖) denote a real Banach space. B(X) and S(X) denote the unit ball and unit sphere of X, respectively. Suppose that ( ) C∗∗ is a ε-separable bounded subset of X∗∗ and C is a closed convex set; ( ) f is a continuous convex function and ∂f(X∗) ⊂ C∗∗; ( ) for any D ⊂ C and weak neighborhood U of origin, there exists a slice S(x∗, C, α) such that S(x∗, D, α)−S(x∗, D, α) ⊂ U. en f is Gâteaux differentiable on a dense Gδ subset G of X∗. We may assume without loss of generality that there exists a subsequence {n} This implies that the sublinear functional σD is nowhere Ĝateaux differentiable, a contradiction, which finishes the proof. By the proof of Theorem 14, we obtain that every closed convex subset of X is weak dentable. Since l1 is separable, by Theorem 19, we obtain that every bounded subset of c0 is weak dentable. By Theorem 14, we obtain that every convex function G of X∗

Gateaux Differentiability and Weakly Exposed Point

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