Example 3

Let G be a topological group, let Y be a uniform space and let $f : G \to Y$ be a uniformly continuous mapping [G being endowed with its left uniformity (Chapter III, § 3, no. 1)]. For each $s \in G$, let $f_s$ be the mapping $x \to f(sx)$ of G into Y. Then the set of mappings $f_s (s \in G)$ is uniformly equicontinuous, since the relation $x^{-1}x' \in V$ is equivalent to $(sx)^{-1}(sx') \in V$.