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 2
The pair $\struct {X, \norm {\, \cdot \, } }$ is a non-Archimedean normed vector space.

Also see

 * Equivalence of Definitions of Non-Archimedean Vector Space Norm