Definition:Non-Archimedean

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


 * $\displaystyle \left\| x + y \right\| \leq \max \left\{ \| x \|, \| y \| \right\}$

for all $x, y \in X$.

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


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

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

An absolute value $|\cdot |$ on a field $k$ is non-Archimedean if:


 * $\displaystyle \left| x + y \right| \leq \max \left\{ | x |, | y | \right\}$

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

Examples

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