Proposition 10

— Let $F$ be a Hausdorff locally convex space over $\mathbf{R}$ such that there exists in the dual $F'$ of $F$ a sequence $(a'_n)$ that is dense for the weak topology $\sigma(F', F)$ (TVS, II, §6, No. 2). In order that two mappings $f, g$ of $X$ into $F$ be equivalent, it is necessary and sufficient that, for every $n$, the numerical functions $\langle f(x), a'_n \rangle$ and $\langle g(x), a'_n \rangle$ be equivalent.

The condition is obviously necessary. Conversely, if it is satisfied, there exists a negligible set $H$ such that, for each $x \notin H$, $\langle f(x), a'_n \rangle = \langle g(x), a'_n \rangle$ for every $n$; this means that the weakly continuous linear forms $z' \mapsto \langle f(x), z' \rangle$ and $z' \mapsto \langle g(x), z' \rangle$ on $F'$ are equal at each of the points $a'_n$, hence are identical by virtue of the hypothesis, which proves that $f(x) = g(x)$ for all $x \notin H$.

Note that the hypothesis of Prop. 10 is applicable in particular when $F$ is a locally convex space that is metrizable and separable¹ (TVS, III, §3, No. 4, Cor. 2 of Prop. 6).