Corollary 1

Let $X$ be a topological (resp. uniform) space, $Y$ a uniform space and $H$ a subset of $\mathscr{F}(X; Y)$. For each $x \in X$, let $\tilde{x}$ denote the mapping $h \to h(x)$ of $H$ into $Y$. Then $H$ is equicontinuous at $x_0$ (resp. uniformly equicontinuous) if and only if the mapping $x \to \tilde{x}$ of $X$ into the uniform space $\mathscr{F}_u(H; Y)$ is continuous at $x_0$ (resp. uniformly continuous).

In particular, if $X$ is compact, every continuous mapping of $X$ into $\mathscr{F}_u(H; Y)$ is uniformly continuous (Chapter II, § 4, no. 1, Theorem 2). Therefore: