A Characterization of Banach Spaces with Separable Duals via Weak Statistical Convergence

Let B be a Banach space. A B-valued sequence Œxk is weakly statistically null provided limn 1 n Ž”k n x Žf xk‘Ž > "•Ž D 0 for all " > 0 and every continuous linear functional f on B. A Banach space is nite 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 l1 is given, and it is also shown that l2 has a “statistical M-basis” which is not a Schauder basis