(Valuations) A point $x$ of $Spec^{an}(R/K)$ is an additive valuation

$$val_{x}:R\to{\bf R}\cup\{+\infty\}$$

extending $val:=val_{K}$, i.e. it is a map satisfying the conditions

• •

  $val_{x}(r+r^{\prime})\leq\max(val_{x}(r),val_{x}(r^{\prime}))$;

• •

  $val_{x}(rr^{\prime})=val_{x}(r)+val_{x}(r^{\prime})$;

• •

  $val_{x}(\lambda)=val_{K}(\lambda)$

for all $r,r^{\prime}\in R$ and all $\lambda\in K$.