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