Equivalence of Definitions of Non-Archimedean Vector Space Norm

Theorem
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 1 implies Definition 2
Let $\norm {\,\cdot\,} : X \to \R_{\ge 0}$ be a norm on a division ring satisfying:

It remains only to show that $\norm {\,\cdot\,}$ satisfies $(\text N 1)$ and $(\text N 2)$.

This follows from the definition of a norm on a division ring.

Definition 2 implies Definition 1
Let $\norm{\,\cdot\,} : X \to \R_{\ge 0}$ satisfy the non-Archimedean norm axioms: $(\text N 1)$, $(\text N 2)$ and $(\text N 4)$.

To show that $\norm{\,\cdot\,}$ is a norm on a division ring satisfying $(\text N 4)$, it remains to show that $\norm{\,\cdot\,}$ satisfies:

Let $x, y \in X$.

, suppose $\norm x \le \norm y$.

From non-Archimedean norm axiom $(\text N 1)$ : Positive Definiteness:
 * $0 \le \norm x$

Then:

The result follows.