Let $X$ be a Banach space and $S\subseteq X$ be a subspace such that the unit sphere of $S$ is weakly compact. If $Y^\*=X$ for some separable Banach space $Y,$ is it true that $S$ is separable?



Weaky compact subset of Banach space with separable predual
Assume that Samya Kumar Ray meant that the unit ball of $S$ is weakly compact. If $S$ is really a sphere, the fact that the answer is "Yes" follows from the argument outlined by Bill Johnson (and the additional assumptions are not needed).


The answer is "yes" anyway. Possibly a bit more straightforward argument than the one of Dirk Werner: the weak$^\*$ topology on the unit ball of $S$ is weaker than the weak topology, so by the well-known result on compacts this implies that on the unit ball of $S$ weak$^\*$ topology is the same as the weak topology. It is well-known and easy to show that bounded sets in the dual to a separable Banach space are weak$^\*$ separable. Thus the unit ball of $S$ is weak separable. It remains to prove the easy statement that Banach spaces whose balls are weakly separable are separable. 


