Characterizations of ball-covering of separable Banach space and application.

In this paper, we first prove that the space (X , ‖ ⋅ ‖) is separable if and only if for every ε ∈ (0 , 1) , there is a dense subset G of X ∗ and a w ∗ -lower semicontinuous norm ‖ ⋅ ‖ 0 of X ∗ so that (1) the norm ‖ ⋅ ‖ 0 is Frechet differentiable at every point of G and d F ‖ x ∗ ‖ 0 ∈ X is a w ∗ -strongly exposed point of B (X ∗ ∗ , ‖ ⋅ ‖ 0) whenever x ∗ ∈ G ; (2) (1 + ε 2 ) ‖ x ∗ ∗ ∗ ‖ 0 ≤ ‖ x ∗ ∗ ∗ ‖ ≤ (1 + ε) ‖ x ∗ ∗ ∗ ‖ 0 for each x ∗ ∗ ∗ ∈ X ∗ ∗ ∗ ; (3) there exists { x i ∗ } i = 1 ∞ ⊂ G such that ball-covering { B (x i ∗ , r i) } i = 1 ∞ of (X ∗ , ‖ ⋅ ‖ 0) is (1 + ε) − 1 -off the origin and S (X ∗ , ‖ ⋅ ‖) ⊂ ∪ i = 1 ∞ 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. In this paper, we first prove that the space is separable if and only if for every , there is a dense subset of and a -lower semicontinuous norm of so that (1) the norm is Frechet differentiable at every point of and is a -strongly exposed point of whenever ; (2) for each ; (3) there exists such that ball-covering of is -off the origin and . Moreover, we also prove that if space is weakly locally uniform convex, then the space is separable if and only if has the ball-covering property. As an application, we get that Orlicz sequence space has the ball-covering property. [ABSTRACT FROM AUTHOR]