The Ball-Covering Property on Dual Spaces and Banach Sequence Spaces

Abstract

In this paper, we prove that (X, p) is separable if and only if there exists a w*-lower semicontinuous norm sequence $\left\{ {{p_n}} \right\}_{n = 1}^\infty$ of (X*, p) such that (1) there exists a dense subset G_n of X* such that p_n is Gâteaux differentiable on G_n and dp_n (G_n) ⊂ X for all n ∊ N; (2) p_n ≤ p and p_n → p uniformly on each bounded subset of X*; (3) for any α ∈ (0, 1), there exists a ball-covering $\left\{ {B\left( {x_{i,n}^*,{r_{i,n}}} \right)} \right\}_{i = 1}^\infty$ of (X*, p_n) such that it is α-off the origin and x ^*_{i, n} ∈ G_n. Moreover, we also prove that if X_i is a Gâteaux differentiability space, then there exist a real number α > 0 and a ball-covering $\mathfrak{B}_i$ of X_i such that $\mathfrak{B}_i$ is α-off the origin if and only if there exist a real number α > 0 and a ball-covering $\mathfrak{B}$ of l^∞ (X_i) such that $\mathfrak{B}$ is α-off the origin.