Proposition 5

Let $G, G'$ be two topological groups endowed with their left uniformities, and let $H$ be a set of homomorphisms of $G$ into $G'$. Then the following conditions are equivalent: a) $H$ is equicontinuous at the identity element $e$ of $G$, b) $H$ is equicontinuous, c) $H$ is uniformly equicontinuous.

It is enough to show that a) implies c). Let $V'$ be a neighbourhood of the identity element $e'$ of $G'$; then, by hypothesis, there is a neighbourhood $V$ of $e$ in $G$ such that $u(V) \subset V'$ for all $u \in H$; since the elements of $H$ are homomorphisms, the relation $x^{-1} y \in V$ implies that we have

In view of the definition of the entourages of the left uniformities of $G$ and $G'$ (Chapter III, § 3, no. 1), the result follows.