= Weakly compact subsets of l-infinity are norm separable
Coordinate evaluations on the <l-infinity sequence space> form a countable separating family, so a <weakly compact set> $K$ is metrizable by <countable separating family metrizes a weakly compact set>. Choose a countable weakly dense subset $D$. Then $K$ lies in the weak closure of $\operatorname{conv}D$, which equals its norm closure by <Mazur theorem>. Rational convex combinations make that norm closure separable, and every subset of a separable metric space is separable. The weakly dense subset itself need not be norm dense in a nonconvex $K$.
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