Proposition 2

The left (resp. right) uniformity of a metrizable group $G$ can be defined by a left-invariant (resp. right-invariant) metric on $G$.

Suppose that the fundamental system $(V_n)$ of neighbourhoods of $e$ consists of symmetric neighbourhoods such that $V_{n+1}^3 \subset V_n$ for each $n$. Then the corresponding entourages $U_n$ of the left uniformity are symmetric entourages such that $U_{n+1} \subset U_n$. The method used in the proof of Proposition 2 of § 1, no. 4 allows us to construct, from the sequence of entourages $(U_n)$, a metric $d$ on $G$ compatible with the left uniformity of $G$; and since for each $z \in G$ the mapping $(x, y) \to (zx, zy)$ leaves each of the $U_n$ invariant, the definition of $d$ shows that it is a left-invariant metric. This method also holds for the right uniformity.

Note that, if the two uniformities of $G$ are distinct, the metric $d$ is not right-invariant, and hence in general $d(x^{-1}, y^{-1}) \neq d(x, y)$.

In particular, if $G$ is a metrizable abelian group, its uniformity is defined by an invariant metric $d$: if $G$ is written additively, we have

We shall often write $|x|$ (or $||x|$) for $d(0, x)$; we have then

The function $|x|$ satisfies the following three conditions:

a) $|-x| = |x|$ for all $x \in G$.

b) $|x + y| \leq |x| + |y|$ for all $x \in G$ and all $y \in G$.

c) $|x| = 0$ if and only if $x = 0$.

Conversely: