Proposition 1

For a mapping $f$ from $K$ to $\mathbf{R}$, satisfying (VA,) and (VAII) to belong to $\mathscr{V}(K)$, it is necessary and sufficient that $f(1 + x)$ be bounded in the set of $x \in K$ such that $f(x) \leq 1$.

If $f$ satisfies $(\mathbf{U}_A)$, then $f(1 + x) \leq A$ if $f(x) \leq 1$. Conversely, suppose that $f(x + 1) \leq A$ for the $x \in K$ such that $f(x) \leq 1$ (which implies that $A \geq f(1) = 1$); then, if $x = 0$ or $y = 0$, condition $(\mathbf{U}_A)$ is fulfilled; if on the other hand $x \neq 0$ and $y \neq 0$, we may assume for example that $f(y) \leq f(x)$, hence, by $(\mathbf{VA}_s)$, $f(yx^{-1}) \leq 1$ and therefore $f(1 + yx^{-1}) \leq A$, which gives, by virtue of $(\mathbf{VA}_{II})$, $f(x + y)f(x)^{-1} \leq A$; whence

If $f$ is an absolute value on $K$, then $f(n.1) \leq n$ by induction on the integer $n > 0$ starting from $(\mathbf{VA}_{III})$; conversely: