# Definition:Non-Archimedean

## Definition

### Non-Archimedean Norm (Vector Space)

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

 $(N4)$ $:$ Ultrametric Inequality: $\displaystyle \forall x, y \in R:$ $\displaystyle \norm {x + y}$ $\displaystyle \le$ $\displaystyle \max \set {\norm x, \norm y}$

### Definition 1

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

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

### 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 $x, y, z \in X$.