Axiom:Non-Archimedean Norm Axioms/Vector Space

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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 a non-Archimedean vector space norm 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:      \(\ds \forall x, y \in X:\)    \(\ds \norm {x + y} \)   \(\ds \le \)   \(\ds \max \set {\norm x, \norm y} \)      

These criteria are called the non-Archimedean norm axioms.

Also see