Definition 6.2 (Norm). Let $(K,\left|\bullet \right|)$ be a non-archimedean valued field, $V$ a vector space over $K$. A normon $V$ is a function $\parallel \bullet \parallel :V\to {ℝ}_{\ge 0}$ satisfying:

• (i) $\parallel x\parallel =0⟺x=0$.
• (ii) $\parallel \lambda x\parallel =\parallel \lambda \parallel \parallel x\parallel$ for all $\lambda \in K$, $x\in V$.
• (iii) $\parallel x+y\parallel \le \max (\parallel x\parallel ,\parallel y\parallel )$ for all $x,y\in V$.