 Metrization of the weak topology on a bounded set

Let $X^{\ast }$ have a norm-dense sequence $({\varphi }_m)$, and let $B\subset X$ be norm-bounded. Thenis a metric on $B$ that induces its weak topology. Density lets convergence against the ${\varphi }_m$ extend uniformly on $B-B$ to convergence against every $\varphi \in X^{\ast }$.