Corollary 5

Let X be a topological space, let Y, Z be two uniform spaces and let H be an equicontinuous set of mappings of Y into Z. If H, $\mathscr{C}(X; Y)$ and $\mathscr{C}(X; Z)$ are endowed with the topology of pointwise convergence, then the mapping $(u, v) \to u \circ v$ of $H \times \mathscr{C}(X; Y)$ into $\mathscr{C}(X; Z)$ is continuous.

We have to show that, for each $x \in X$, the mapping $(u, v) \to u(v(x))$ of $H \times \mathscr{C}(X; Y)$ into Z is continuous. Now $v \to v(x)$ is continuous on H (§ 1, no. 2, Remark 6), and it follows from Corollary 4 that $(u, y) \to u(y)$ is a continuous mapping of $H \times Y$ into Z; since $(u, v) \to u(v(x))$ is the composition of $(u, y) \to u(y)$ and $(u, v) \to (u, v(x))$, the result is proved.

The following proposition and its corollary are the analogues of Corollaries 3 and 4 of Proposition 1 for uniformly equicontinuous sets of mappings: