Corollary 3

Let $H$ be a subset of $\mathscr{F}(X; Y)$ such that, for each

the closure of $H(x)$ in $Y$ is a complete subspace of $Y$. Then the closure $\overline{H}$ of $H$ in $\mathscr{F}_{\mathscr{G}}(X; Y)$ is a complete subspace.

Let $\Phi$ be a Cauchy filter on $\overline{H}$, and define a mapping $v : X \to Y$ as follows. If $x \in B$, $\Phi(x)$ is a Cauchy filter on $\overline{H(x)}$ (no. 2, Remark 6), hence by hypothesis it has at least one limit point; take $v(x)$ to be one of these limits. If $x \notin B$, take $v(x)$ to be any point of $Y$. With this definition of $v$, it is clear that $\Phi$ converges pointwise to $v$ in $B$, and $v$ is therefore a limit of $\Phi$ in $\mathscr{F}_{\mathscr{G}}(X; Y)$ by Proposition 5.

In particular, if $Y$ is complete, the hypothesis of Corollary 3 of Proposition 5 is satisfied for every $H \subset \mathscr{F}(X; Y)$; hence: