Definition 2

An absolute value on a division ring K is a mapping $x \to |x|$ of K into $\mathbf{R}_+$ which satisfies the following conditions:

(VM_I) $|x| = 0$ if and only if $x = 0$. (VM_II) $|xy| = |x| \cdot |y|$ for all $x, y$ in K. (VM_III) $|x + y| \leq |x| + |y|$ for all $x, y$ in K.

By (VM_II) we have $|x| = |1| \cdot |x|$, and since by (VM_I) there is at least one x such that $|x| \neq 0$, we have $|1| = 1$; it follows that $1 = |1|^2$, hence $|1| = 1$ and consequently

therefore $|x - y| \leq |x| + |y|$ for all $x, y$ in K. We can therefore say that $d(x, y) = |x - y|$ is an invariant metric on the additive group K, and that the mapping $x \to |x|$ is a homomorphism of the multiplicative group $K^*$ of non-zero elements of $K$ into the multiplicative group $\mathbf{R}_+^*$ of real numbers $> 0$.

The invariant metric $|x - y|$ defines a metric space topology on $K$, compatible with its additive group structure (no. 1, Proposition 3); but, moreover, this topology is compatible with the division ring structure of $K$. For the continuity of $xy$ on $K \times K$ follows from the relation

which gives

Likewise, the continuity of $x^{-1}$ at every point $x_0 \neq 0$ follows from the identity $x^{-1} - x_0^{-1} = x^{-1}(x_0 - x)x_0^{-1}$, which gives, by (VM_{II}),

now if $\varepsilon > 0$ is such that $\varepsilon < |x_0|$, the relation $|x - x_0| \leq \varepsilon$ implies $|x| \geq |x_0| - \varepsilon$, whence $|x^{-1} - x_0^{-1}| \leq \varepsilon / |x_0|(|x_0| - \varepsilon)$; and this establishes the continuity of $x^{-1}$ at the point $x_0$.