Axiom:Non-Archimedean Norm Axioms/Vector Space

Definition

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

• Definition:Non-Archimedean Vector Space Norm