# Category:Axioms/Non-Archimedean Norm Axioms

Jump to navigation Jump to search

This category contains axioms related to Non-Archimedean Norm Axioms.

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

## Pages in category "Axioms/Non-Archimedean Norm Axioms"

The following 3 pages are in this category, out of 3 total.