Let $E$ be a Banach space. Suppose that $E'$ is weakly-* sequentially separable, that is, that there exists a countable $D \subset E'$ s.t. every $x' \in E'$ is a limit point of a sequence in $D$. Does it follow that $E$ is separable?
This question arises from a conversation with a friend of mine. He thinks this is true and plans to use it to prove separability of $C(K)$ for a metrizable and compact Hausdorff $K$. I'm not so sure this can work, though. Of course, if $E'$ is norm separable then $E$ is separable, but we are talking about a much weaker topology here.