Definition Let $(F,|\cdot|)$ be a field equipped with an absolute value. A normed vector space over $F$ is a pair $(V,\|\cdot\|)$ consisting of an $F$-vector space $V$ and a map $\|\cdot\| : V → \mathbb{R}$ satisfying
- $\|v\|\geq 0$ with equality if and only if $v = 0$,
- $\|cv\| = |c|\cdot\|v\|$ for all $c\in F$ and $v\in V$,
- $\|v + w\| \leq \|v\| + \|w\|$ for all $v,w \in V$.
How to prove that if $F$ is non-archimedean then there is a constant $C > 0$ (depending on the normed vector space) such that $\|v+w\| \leq C\max(\|v\|,\|w\|)$ for all $v,w \in V$.