Countable separating family metrizes a weakly compact set ($d(x,y)={\sum }_{n\ge 1}{2}^{-n}\frac{|f_n(x-y)|}{1+|f_n(x-y)|}$)

If a countable family in $X^{\ast }$ 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.