\begin{definition}[Definition:Norm/Vector Space]
Let $\struct {R, +, \circ}$ be a division ring with norm $\norm {\,\cdot\,}_R$.
Let $V$ be a vector space over $R$, with zero $0_V$.
A '''norm''' on $V$ is a map from $V$ to the nonnegative reals:
:$\norm{\,\cdot\,}: V \to \R_{\ge 0}$
satisfying the '''(vector space) norm axioms''':
{{Axiom:Norm Axioms (Vector Space)}}
\end{definition}