Theorem 2 (Ascoli)

Let $X$ be a topological (resp. uniform) space, let $\mathscr{S}$ be a covering of $X$, let $Y$ be a uniform space and $H$ a set of mappings of $X$ into $Y$ such that, for each $A \in \mathscr{S}$ and each $u \in H$, the restriction of $u$ to $A$ is continuous (resp. uniformly continuous). Then, for $H$ to be precompact with respect to the uniformity of $\mathscr{S}$-convergence, it is necessary in all cases and also sufficient if the sets $A \in \mathscr{S}$ are compact (resp. precompact) that the following conditions should be satisfied:

a) For each $A \in \mathscr{S}$, the set $H|A \subset \mathscr{F}(A; Y)$ of restrictions to $A$ of functions of $H$ is equicontinuous (resp. uniformly equicontinuous).

b) For each $x \in X$, the set $H(x) \subset Y$ of points $u(x)$ ($u \in H$) is precompact.

1) Let us show first that conditions a) and b) are necessary. We know (§ 1, no. 2, Remark 6) that the mapping $u \to u(x)$ of $\mathscr{F}_{\mathscr{S}}(X; Y)$ into $Y$ is uniformly continuous; hence, if $H$ is precompact, so is $H(x)$ (Chapter II, § 4, no. 2, Proposition 2), which proves b). To prove a), consider a set $A \in \mathscr{S}$, a point $x_0 \in A$ and an entourage $V$ of $Y$; since $H$ is precompact it can be covered by a finite number of $W(A, V)$-small sets; in other words there is a finite sequence $(u_i)$ of elements of $H$ such that, for each $u \in H$, we have

for at least one index $i$.

Since each of the $u_i|A$ is continuous at $x_0$ (resp. uniformly continuous) there is a neighbourhood $U_i$ of $x_0$ in $A$ (resp. an entourage $M_i$ of $A$) such that

(resp. such that

Let $U$ (resp. $M$) be the intersection of the $U_i$ (resp. $M_i$); it is a neighbourhood of $x_0$ in $A$ (resp. an entourage of $A$). For each $u \in H$ there is an index $i$ for which (4) holds; writing condition (4) for $x_0$ and for $x$ (resp. for $x'$ and $x''$) and taking account of (5) [resp. (6)], we see immediately that the relation $x \in U$ [resp. $(x', x'') \in M$] implies $(u(x),\ u(x_0)) \in \overset{\circ}{V}$ [resp. $(u(x'),\ u(x'')) \in \overset{\circ}{V}$], for each $u \in H$; and this establishes a).

2) Now let us show that the conditions a) and b) are sufficient if the sets $A \in \mathscr{S}$ are compact (resp. precompact). Condition b) implies that $H$ is precompact with respect to the uniformity of pointwise convergence (Chapter II, § 4, no. 2, Proposition 3). But it follows from condition a) and Theorem 1 of no. 4 that on $H|A$ the uniformity of pointwise convergence in $A$ coincides with the uniformity of uniform convergence in $A$; hence $H|A$ is precompact in $\mathscr{F}_u(A; Y)$, which implies that $H$ is precompact with respect to the uniformity of $\mathscr{S}$-convergence (§ 1, no. 2).

Note that condition b) of Theorem 2 is automatically satisfied if $Y$ is a precompact space.