is a special field that contains Laurent series without essential singularities. We are going to explain how is constructed a Berkovich space over $\LL$. For simplicity we assume in this section that $\LL$ is the set of these series. Let us define a norm on this space. Let $$c:=\sum_{k\geq k_0}c_{j} t^{k}\in \LL\quad \text{with}\quad k_0\in\Z\text{ and }c_{k_0}\neq0.$$

We define $$|c|_\LL:=exp({-k_0}).$$ Setting $|0|_\LL=0$, we assure that $|.|_\LL$ is a norm on $\LL$. Moreover, it satisfies an inequality stronger then the usual triangular inequality, called the ultrametric or non-Archimedean inequality and that defines $\LL$ to be a non-Archimedean field: $$\forall a,b\in\LL, |a+b|_\LL\leq \rm{max}(|a|_\LL,|b|_\LL).$$

This inequality, is very strange for people who are not used to it. For example I have here to say that this paper design by “closed” (resp. “open”) ball the ball define by a large (resp. strict) inequality because both of them are open and closed for this topology. Also, if we denote by $B(z,r)$ the closed ball of center $z\in\LL$ and radius $r\geq0$, we have the surprising following lemma.

\[ball\] Given $z,z'\in\LL$ and $r\geq0$, we have $$z\in B(z',r)\implies B(z,r)=B(z',r).$$

Suppose that $|z-z'|\leq r$. Then $$x\in B(z,r)\implies |x-z'|_\LL\leq \max (|x-z|_\LL,|z-z'|_\LL)\leq r\implies x\in B(z',r).$$

This means that every element in a 