Axiom:Non-Archimedean Norm Axioms/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 $0_X$.

Let $\norm {\, \cdot \,}: X \to \R_{\ge 0}$ be a mapping from $X$ to the non-negative reals.

$\norm {\, \cdot \,}$ is a non-Archimedean vector space norm $\norm {\, \cdot \,}$ satisfies the following contitions:

These criteria are called the non-Archimedean norm axioms.

Also see

 * Definition:Non-Archimedean Vector Space Norm