Definition:Non-Archimedean

Definition

Non-Archimedean Norm

A norm $\left\Vert \cdot \right\Vert$ on a space $X$ is non-Archimedean if it satisfies the ultrametric inequality:

$\left\Vert {x + y} \right\Vert \leq \max \left\{ {\left\Vert {x} \right\Vert, \left\Vert {y} \right\Vert} \right\}$

for all $x, y \in X$.

Non-Archimedean Metric

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

$d \left({x, y}\right) \leq \max \left\{ {d \left({x, z}\right), d \left({y, z}\right)} \right\}$

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

Non-Archimedean Absolute Value

An absolute value $\left\vert {\cdot}\right\vert$ on a valued field $k$ is non-Archimedean if:

$\left\vert{x + y} \right\vert \leq \max \left\{ {\left\vert{x}\right\vert, \left\vert{y}\right\vert} \right\}$

Archimedean

A norm, metric or absolute value) is Archimedean if it is not non-Archimedean.

Examples

• The $p$-adic metric is non-Archimedean.