Definition:Non-Archimedean/Norm (Vector Space)/Definition 1

Definition

Let $X$ be a 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:

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

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

\(\ds \le \)

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

Also see

• Equivalence of Definitions of Non-Archimedean Vector Space Norm