On Banach spaces whose dual balls are not weak sequentially compact

Theorem 1. Let X be a Banach space. (a) If X * has a closed subspace in which no normalized sequence converges weak* to zero, then l 1 is isomorphic to a subspace of X. (b) If X * contains a bounded sequence which has no weak* convergent subsequence, then X contains a separable subspace whose dual is not separable. 1997 The Weizmann Science Press of Israel.

On Banach spaces whose dual balls are not weak sequentially compact Academic Article