Definition:Non-Archimedean/Norm (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 $0_X$.

Definition 1

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

\((\text N 4)\)

$:$

Ultrametric Inequality:

\(\ds \forall x, y \in X:\)

\(\ds \norm {x + y} \)

\(\ds \le \)

\(\ds \max \set {\norm x, \norm y} \)

Definition 2

A non-Archimedean norm on $X$ is a mapping from $X$ to the non-negative reals:

$\norm {\, \cdot \,}: X \to \R_{\ge 0}$

satisfying the non-Archimedean norm axioms:

\((\text N 1)\)	$:$		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:	\(\ds \forall x, y \in X:\)		\(\ds \norm {x + y} \)	\(\ds \le \)	\(\ds \max \set {\norm x, \norm y} \)

The pair $\struct {X, \norm {\, \cdot \, } }$ is a non-Archimedean normed vector space.

Archimedean

A norm $\norm {\,\cdot\,} $ on a vector space $X$ is Archimedean if and only if it is not non-Archimedean.

Also see

• Equivalence of Definitions of Non-Archimedean Vector Space Norm