Proposition 3

Let $f$ be a mapping of $K$ to $\mathbf{R}_+$. The following properties are equivalent: (a) $f$ is an ultrametric absolute value. (b) There exists a valuation $v$ on $K$ with values in $\mathbf{R}$ and a real number $a$ such that $0 < a < 1$ and $f = a''$. (c) $f$ belongs to $\mathscr{V}(K)$ and $f(n.1) \leq 1$ for every integer $n > 0$. (d) For all $s > 0$, $f^s$ is an absolute value.

For every real number $c$ such that $0 < c < 1$, the mapping $t \mapsto c^t$ is an isomorphism of the ordered group $\mathbf{R}$ (with the opposite ordering to the usual ordering) on the ordered group $\mathbf{R}_+^*$; this shows the equivalence of (a) and (b). Clearly (a) implies (c); (c) implies (d), for we deduce from (c) that

for every integer $n > 0$ and Proposition 2 of no. 1 shows that $f''$ is an absolute value. Finally (d) implies (a): for iff" is an absolute value, it satisfies $(\mathbf{U}_2)$ and hence $f$ satisfies $(\mathbf{U}_{2^{1/s}})$ for all $s > 0$ and therefore also $(\mathbf{U}_1)$, letting $s$ tend to $+\infty$.