Proposition 1.4. Assuming that:

Then the following are equivalent:

(i) $| ∙ |$ and ${| ∙ |}^{'}$ are equivalent.
(ii) $| x | < 1 ⟺ {| ∙ |}^{'} < 1$ for all $x \in K$ .
(iii) There exists $c \in ℝ_{> 0}$ such that ${| x |}^{c} = {| ∙ |}^{'}$ for all $x \in K$ .