Proposition 3

Let $G$ be an abelian group, written additively, and let $x \to |x|$ be a mapping $G \to \mathbf{R}$ which satisfies conditions a), b) and c) above. Then the function $d(x, y) = |x - y|$ is an invariant metric on $G$; the topology $\mathscr{T}$ which it defines on $G$ is compatible with the group structure of $G$, and the uniformity defined by $d$ is the same as the uniformity of the topological group obtained by endowing $G$ with the topology $\mathscr{T}$.

The function $d(x, y)$ is a metric on $G$, for the relation $d(x, y) = 0$ is equivalent to $x = y$ by c); we have $d(x, y) = d(y, x)$ by a); and

by b). Moreover, $d$ is invariant, since $(x + z) - (y + z) = x - y$. For each real number $x > 0$, let $V_x$ be the set of all $x \in G$ such that $|x| < x$; then the $V_x$ form a fundamental system $\mathfrak{S}$ of neighbourhoods of $0$ for the topology $\mathscr{T}$, and since $d$ is invariant, $a + \mathfrak{S}$ is a fundamental system of neighbourhoods of $a$ for the topology $\mathscr{T}$, for each $a \in G$. By a), the $V_x$ are symmetric, and by b), we have $V_a + V_x \subset V_{2a}$; hence the topology $\mathscr{T}$ is compatible with the group structure of $G$ (Chapter III, § 1, no. 2). The last part of the proposition follows immediately.

Conditions a', b' and c' are equivalent to c) together with the condition

For a' and b' clearly imply b'); conversely, taking $x = 0$ in b') and using c), we see that $|-y| \leq |y|$; replacing $y$ by $-y$ it follows that $|-y| = |y|$, which is a); replacing $y$ by $-y$ in b') we then get b').