A characterization of Banach spaces with separable duals via weak statistical convergence

Let B be a Banach space. A B-valued sequence (x(k)) is weakly statistically null provided lim(n) 1/n \{k less than or equal to n : \ f(x(k))\ > epsilon}\ = 0 for all epsilon > 0 and every continuous linear functional f on B. A Banach space is finite dimensional if and only if every weakly statistically null B-valued sequence has a bounded subsequence. If B is separable, B* is separable if and only if every bounded weakly statistically null B-valued sequence contains a large weakly null sequence. A characterization of spaces containing an isomorphic copy of l(1) is given, and it is also shown that l(2) has a "statistical M-basis" which is not a Schauder basis. (C) 2000 Academic Press.