Let $X$ be a Banach space, and let $B$ be the closed unit ball of $X^*$, equipped with the weak-* topology. Alaoglu's theorem says that $B$ is compact. If $X$ is separable, then $B$ is metrizable, and in particular it is sequentially compact.
What about the converse? If $B$ is sequentially compact, must $X$ be separable?
This question was inspired by this one.