= Countable separating family metrizes a weakly compact set
{title2=$d(x,y)=\sum_{n\ge1}2^{-n}\frac{|f_n(x-y)|}{1+|f_n(x-y)|}$}
If a countable family in $X^*$ separates points of $X$, the displayed metric induces the <weak topology> on every <weakly compact set> $K$. The coordinate map into the countable product of scalar lines is continuous and injective; a continuous injection from a compact space into a Hausdorff space is a homeomorphism onto its image. Equivalently, uniform convergence of the metric series makes the identity from weak $K$ to metric $K$ continuous. Compactness is essential: point separation alone need not generate the weak topology on an arbitrary bounded set.
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