Definition:Non-Archimedean

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Definition

Non-Archimedean Norm (Vector Space)

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

\((\text N 4)\)   $:$   Ultrametric Inequality:      \(\ds \forall x, y \in X:\)    \(\ds \norm {x + y} \)   \(\ds \le \)   \(\ds \max \set {\norm x, \norm y} \)      


Non-Archimedean Norm (Division Ring)

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

\((\text N 4)\)   $:$   Ultrametric Inequality:      \(\ds \forall x, y \in R:\)    \(\ds \norm {x + y} \)   \(\ds \le \)   \(\ds \max \set {\norm x, \norm y} \)      


Non-Archimedean Metric

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 $x, y, z \in X$.