Corollary

Under the hypotheses of Theorem 1, the closure $\overline{H}$ of $H$ in $\mathscr{F}(X; Y)$ with respect to the topology of pointwise convergence is the same as the closure of $H$ in $\mathscr{C}(X; Y)$ with respect to the topology of compact (resp. precompact) convergence.

For the set $\overline{H}$ is equicontinuous (resp. uniformly equicontinuous) by Proposition 6 of no. 3, and hence is contained in $\mathscr{C}(X; Y)$; the result follows immediately from the fact that, on $\overline{H}$, the two topologies under consideration are the same, by virtue of Theorem 1.