Definition:Non-Archimedean

Definition

A norm $\norm {\,\cdot\,} $ on $X$ is non-Archimedean if and only if $\norm {\, \cdot \,}$ satisfies the axiom:

$(\text N 4)$

$:$

Ultrametric Inequality:

$\ds \forall x, y \in X:$

$\ds \norm {x + y} $

$\ds \le $

$\ds \max \set {\norm x, \norm y} $

A norm $\norm {\, \cdot \,}$ on $R$ is non-Archimedean if and only if $\norm {\, \cdot \,}$ satisfies the axiom:

$(\text N 4)$

$:$

Ultrametric Inequality:

$\ds \forall x, y \in R:$

$\ds \norm {x + y} $

$\ds \le $

$\ds \max \set {\norm x, \norm y} $

A metric $d$ on a metric space $X$ is non-Archimedean if and only if:

$\map d {x, y} \le \max \set {\map d {x, z}, \map d {y, z} }$

for all $x, y, z \in X$.