关键词: Ball-covering property, separable space, G$\mathrm{\hat{a}}$teaux differentiable point, weak$^{*}$ exposed point

Abstract: In this paper, we prove that $(X,p)$ is separable if and only if there exists a $w^{*}$-lower semicontinuous norm sequence $\{ {p_n}\} _{n = 1}^\infty$ of $(X^{*},p)$ such that (1) there exists a dense subset $G_{n}$ of $X^{*}$ such that $p_{n}$ is G$\mathrm{\hat{a}}$teaux differentiable on $G_{n}$ and $dp_{n}(G_{n})\subset X$ for all $n\in N$; (2) $p_n \leq p$ and $p_n \to p$ uniformly on each bounded subset of $X^{*}$; (3) for any $\alpha\in(0,1)$, there exists a ball-covering $\{ B({x_{i,n}^{*}},{r_{i,n}})\} _{i = 1}^\infty$ of $(X^{*},p_{n})$ such that it is $\alpha$-off the origin and ${x_{i,n}^{*}}\in G_{n}$. Moreover, we also prove that if $X_{i}$ is a G$\mathrm{\hat{a}}$teaux differentiability space, then there exist a real number $\alpha > 0$ and a ball-covering $\mathfrak{B_{i}}$ of $X_{i}$ such that $\mathfrak{B_{i}}$ is $\alpha$-off the origin if and only if there exist a real number $\alpha > 0$ and a ball-covering $\mathfrak{B}$ of ${l^\infty }({X_i})$ such that $\mathfrak{B}$ is $\alpha$-off the origin.

Key words: Ball-covering property, separable space, G$\mathrm{\hat{a}}$teaux differentiable point, weak$^{*}$ exposed point