Characterizations of ball-covering of separable Banach space and application

In this paper, we first prove that the space (X, parallel to center dot parallel to) is separable if and only if for every epsilon is an element of (0, 1), there is a dense subset G of X* and a w*-lower semicontinuous norm parallel to center dot parallel to(0) of X* so that (1) the norm parallel to center dot parallel to(0) is Frechet differentiable at every point of G and dF parallel to x*parallel to(0) is an element of X is a w*-strongly exposed point of B(X**, parallel to center dot parallel to(0)) whenever x* is an element of G; (2) 1 + epsilon(2)) parallel to x***parallel to(0) <= parallel to x***parallel to <= (1 + epsilon) parallel to x***parallel to(0) for each x*** is an element of X***; (3) there exists {x(i)*}(i=1)(infinity) subset of G such that ball-covering {B(x(i)*, r(i))}(i=1)(infinity) of (X*, parallel to center dot parallel to(0)) is (1 + epsilon)(-1)-off the origin and S (X*, parallel to center dot parallel to ) subset of boolean OR B-infinity(i=1)(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.