Metrization of the weak topology on a bounded set
= Metrization of the weak topology on a bounded set
Let $X^*$ have a norm-dense sequence $(\phi_m)$, and let $B\subset X$ be norm-bounded. Then
$$
d(x,y)=\sum_{m=1}^{\infty}2^{-m}
\frac{|\phi_m(x-y)|}{1+|\phi_m(x-y)|}
$$
is a metric on $B$ that induces its weak topology. Density lets convergence against the $\phi_m$ extend uniformly on $B-B$ to convergence against every $\phi\in X^*$.