Weak-star metrizability criterion for a dual ball

The dual unit ball $B_{X^{\ast }}$ is weak-star metrizable if and only if $X$ is separable. For the converse direction, Banach-Alaoglu theorem makes a metrizable dual ball a compact metric space, and the evaluation map embeds $X$ isometrically into the separable space $C(B_{X^{\ast }})$.