# Axiom:Non-Archimedean Norm Axioms

## Definition

### Non-Archimedean Norm Axioms (Division Ring)

Let $\struct {R, +, \circ}$ be a division ring whose zero is $0_R$.

Let $\norm {\, \cdot \,}: R \to \R_{\ge 0}$ be a non-Archimedean norm on $R$.

The non-Archimedean norm axioms are the conditions on $\norm {\, \cdot \,}$ which are satisfied for all elements of $R$ in order for $\norm {\, \cdot \,}$ to be a non-Archimedean norm:

 $(\text N 1)$ $:$ Positive Definiteness: $\ds \forall x \in R:$ $\ds \norm x = 0$ $\ds \iff$ $\ds x = 0_R$ $(\text N 2)$ $:$ Multiplicativity: $\ds \forall x, y \in R:$ $\ds \norm {x \circ y}$ $\ds =$ $\ds \norm x \times \norm y$ $(\text N 4)$ $:$ Ultrametric Inequality: $\ds \forall x, y \in R:$ $\ds \norm {x + y}$ $\ds \le$ $\ds \max \set {\norm x, \norm y}$

### Non-Archimedean Norm Axioms (Vector Space)

Let $\struct {R, +, \circ}$ be a division ring with norm $\norm {\,\cdot\,}_R$.

Let $X$ be a vector space over $R$, with zero $0_X$.

Let $\norm {\, \cdot \,}: X \to \R_{\ge 0}$ be a mapping from $X$ to the non-negative reals.

$\norm {\, \cdot \,}$ is satisfies the non-Archimedean norm axioms if and only if $\norm {\, \cdot \,}$ satisfies the following contitions:

 $(\text N 1)$ $:$ Positive Definiteness: $\ds \forall x \in X:$ $\ds \norm x = 0$ $\ds \iff$ $\ds x = 0_R$ $(\text N 2)$ $:$ Positive Homogeneity: $\ds \forall x \in X, \lambda \in R:$ $\ds \norm {\lambda x}$ $\ds =$ $\ds \norm {\lambda}_R \times \norm x$ $(\text N 4)$ $:$ Ultrametric Inequality: $\ds \forall x, y \in X:$ $\ds \norm {x + y}$ $\ds \le$ $\ds \max \set {\norm x, \norm y}$