\begin{definition}[Definition:Non-Archimedean/Norm (Vector Space)]
A norm $\norm {\,\cdot\,} $ on a vector space $X$ is '''non-Archimedean''' {{iff}} $\norm {\, \cdot \,}$ satisfies the axiom:
{{begin-axiom}}
{{axiom | n = N4
        | lc= Ultrametric Inequality:
        | q = \forall x, y \in R
        | ml= \norm {x + y}
        | mo= \le
        | mr= \max \set {\norm x, \norm y}
}}
{{end-axiom}}
The pair $\struct {X, \norm {\, \cdot \, } }$ is a '''non-Archimedean normed vector space'''.
\end{definition}