Definition:Non-Archimedean

From ProofWiki
Jump to navigation Jump to search

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


Non-Archimedean Norm (Division Ring)

Definition 1

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

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


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 $x, y, z \in X$.