Definition:Non-Archimedean/Norm (Vector Space)/Definition 2

Definition
Let $\struct {R, +, \circ}$ be a division ring with norm $\norm {\,\cdot\,}_R$.

Let $X$ be a vector space over $R$, with zero $0_X$.

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:

Also see

 * Equivalence of Definitions of Non-Archimedean Vector Space Norm