= Weak-ball metrizability requires a norm-separable dual
{title2=$B_E\text{ weakly metrizable}\iff E'\text{ norm separable}$}
If the relative <weak topology> on the unit ball of a <normed vector space> is first countable at zero, choose a countable collection of finite families of <continuous linear functionals> defining a local base. For any $f\in E'$ and $\epsilon>0$, one family $F$ has common kernel $M$ on which $\|f|_M\|\leq\epsilon$. Extend that restriction to $h\in E'$ with $\|h\|\leq\epsilon$ using the <Hahn-Banach theorem>. Then $f-h$ vanishes on $M$, so it belongs to $\operatorname{span}F$ by finite-dimensional factorization. Consequently the countable union of these families has norm-dense linear span in $E'$. The converse is the <metrization of the weak topology on a bounded set> construction from a norm-dense sequence in $E'$. In particular, separability of $E$ alone does not suffice: $\ell^1$ has nonseparable dual $\ell^\infty$.
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