Weak-star metrizability of the dual ball

If $X$ is separable, the weak-star topology on $B_{X^{\ast }}$ is metrizable. For a dense sequence $(x_n)$ in $B_X$, a compatible metric is obtained by summing bounded multiples of $2^{-n}|f(x_n)-g(x_n)|$.

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 $C(B_{X^{\ast }})$.