Characterizations of ball-covering of separable Banach space and application

Abstract

<abstract><p>In this paper, we first prove that the space $ (X, \|\cdot\|) $ is separable if and only if for every $ \varepsilon\in $ $ (0, 1) $, there is a dense subset $ G $ of $ X^{*} $ and a $ w^{*} $-lower semicontinuous norm $ \|\cdot\|_{0} $ of $ X^{*} $ so that (1) the norm $ \|\cdot\|_{0} $ is Frechet differentiable at every point of $ G $ and $ d_{F}\|x^{*}\|_{0}\in X $ is a $ w^{*} $-strongly exposed point of $ B(X^{**}, \|\cdot\|_{0}) $ whenever $ x^{*}\in G $; (2) $ \left(1+\varepsilon^{2}\right) {\left\| {{x^{***}}} \right\|_0} \le \left\| {{x^{***}}} \right\| \le \left(1 + \varepsilon \right){\left\| {{x^{***}}} \right\|_0} $ for each $ x^{***}\in X^{***} $; (3) there exists $ \{x_{i}^{*}\}_{i = 1}^{\infty}\subset G $ such that ball-covering $ \{ B({x_{i}^{*}}, {r_i})\} _{i = 1}^\infty $ of $ (X^{*}, \|\cdot\|_{0}) $ is $ (1+\varepsilon)^{-1} $-off the origin and $ S(X^{*}, \|\cdot\|)\subset \cup_{i = 1}^{\infty}B({x_{i}^{*}}, {r_i}) $. Moreover, we also prove that if space $ X $ is weakly locally uniform convex, then the space $ X $ is separable if and only if $ X^{*} $ has the ball-covering property. As an application, we get that Orlicz sequence space $ l_{M} $ has the ball-covering property.</p></abstract>