Corollary 2

Let F be a locally convex space that is metrizable and separable, and let $F'$ be its dual. For a mapping f of X into F to be measurable, it is necessary and sufficient that for every $a' \in F'$, the scalar function $x \mapsto \langle f(x), a' \rangle$ be measurable.

We may regard F as a subspace of a countable product $\prod_n E_n$ of Banach spaces (TVS, II, §4, No. 3), and we can suppose that $\operatorname{pr}_n(F)$ is dense in $E_n$, which is therefore separable. For every $n$, the mapping $\operatorname{pr}_n \circ f$ is then measurable by Prop. 10, therefore $f$ is measurable by No. 3, Th. 1.