Exercise 4, § 2

Let X, Y, Z be three uniform spaces. a) Let H be a uniformly equicontinuous subset of $\mathscr{C}(Y; Z)$. If H, $\mathscr{C}(X; Y)$ and $\mathscr{C}(X; Z)$ are endowed with the uniformity of uniform convergence, show that the mapping $(u, v) \to u \circ v$ of $H \times \mathscr{C}(X; Y)$ into $\mathscr{C}(X; Z)$ is uniformly continuous. b) Let K be a uniformly equicontinuous subset of $\mathscr{C}(X; Y)$, and let L be a uniformly equicontinuous subset of $\mathscr{C}(Y; Z)$. Show that the set of mappings $v \circ u$, where $u$ runs through K and $v$ runs through L, is a uniformly equicontinuous subset of $\mathscr{C}(X; Z)$.