We will introduce the notion of Berkovich's analytic spaces, following \cite{Berkovich1}. Berkovich's analytic space is one of the nonarchimedean analogues of the complex analytic space.
Let $(K,\left|\cdot\right|)$ be a valued field. Recall that a \textit{seminorm} on a $K$-vector space $V$ is a map $\|\cdot\|:V\rightarrow\mathbb{R}_{\geq0}$ such that $\|0\|=0$, $\|\lambda x\|=\left|\lambda\right|\|x\|$, and $\|x+y\|\leq\|x\|+\|y\|$ for $x,y\in V$ and $\lambda\in K$.
We first generalize this notion to the case of abelian groups. Let $A$ be an abelian group.
Definition 6.1. A seminorm on $A$ is a map $\|\cdot\| : A\rightarrow \mathbb{R}_{\geq0}$ such that $\|0\| = 0$, $\|-f\|=\|f\|$ , and $\|f+g\|\leq \|f\| + \|g\|$ for all $f,g\in A$. A seminorm is non-archimedean if it satisfies $\|f+g\|\leq \max\{\|f\| , \|g\|\}$.
Remark 6.2. The conditions $\|-f\|=\|f\|$ , and $\|f+g\|\leq \|f\| + \|g\|$ can be equivalently written as $\|f-g\|\leq \|f\|+\|g\|$ for all $f,g\in A$.
Each seminorm $\|\cdot\|$ induces a topology on $A$. Since we have a notion of open balls, a subset $U\subset A$ is open if for every point in $U$, there exists some open ball that contained in $U$.
This topology is Hausdorff if and only if $\|\cdot\|$ is a norm, i.e., $\|f\| = 0 \Leftrightarrow f=0$. In fact, if $\|\cdot\|$ is not a norm, then there exists some $a\in A$ such that $\|a\| = \|a-0\| = 0$, which implies that the points $0\neq a$ do not have disjoint neighborhood.
In terms of the seminorm, we can construct completion of $A$ in the usual sense. A sequence $(x_{n})_{n\in\mathbb N}$ in $A$ is a \textit{Cauchy sequence} if for every $\varepsilon>0$, $\|x_{n}-x_{m}\|<\varepsilon$ when $n,m$ sufficiently large. We define the addition of two Cauchy sequences to be termwise addition. For any two Cauchy sequences $x=(x_{n})_{n\in\mathbb N}$ and $y=(y_{n})_{n\in\mathbb N}$ in $A$, we define a seminorm as follows
$$\|x-y\|_{0}:=\lim_{n\to\infty}\|x_{n}-y_{n}\|.$$
Then we say that two Cauchy sequences $x=(x_{n})_{n\in\mathbb N}$ and $y=(y_{n})_{n\in\mathbb N}$ in $A$ are \textit{equivalent} if $\|x-y\|_{0}=0$. We denote the set of equivalence classes of Cauchy sequences by $\widehat{A}$. And we define the seminorm of an equivalence class $x$ represented by $(x_{n})_{n\in\mathbb N}$ of Cauchy sequences by
$$ \|x\|_{1}:=\lim_{n\to\infty}\|x_{n}\|. $$
The limit exists since $(\|x_{n}\|)_{n\in\mathbb N}$ forms a Cauchy sequence in $\mathbb{R}$. Let $x,y\in\widehat{A}$ which are represented by $x=(x_{n})_{n\in\mathbb N}$ and $y=(y_{n})_{n\in\mathbb N}$ respectively. We define the \textit{addition} of $x$ and $y$ to be the equivalence class $x+y$ represented by $(x_{n}+y_{n})_{n\in\mathbb N}$. So $\widehat{A}$ forms a group under the addition defined before.
Next, we show that $\widehat{A}$ is complete. Let $(x_{n})_{n\in\mathbb N}$ be a Cauchy sequence in $\widehat{A}$. Assume that $(x_{n})_{n\in\mathbb N}$ has no limit in $\widehat{A}$. Then for every $a\in\widehat{A}$ and every integer $N>0$, there exists $\varepsilon>0$ such that $\|x_{v}-a\|_{2}\geq\varepsilon$ for all $v>N$. Choose representative Cauchy sequences $(x_{v,i})_{i\in\mathbb N}$ and $(a_{i})_{i\in\mathbb N}$, then we have $\lim_{i\to\infty}\|x_{v,i}-a_{i}\|\geq\varepsilon$. Since
$$ \|x_{n,i}-a_{i}\|-\|x_{n,i'}-a_{i'}\|\leq\|x_{n,i}-x_{n,i'}+a_{i'}-a_{i}\|\leq\|x_{n,i}-x_{n,i'}\|+\|a_{i}-a_{i'}\|, $$
the limit $\lim_{i\to\infty}\|x_{v,i}-a_{i}\|$ exists. This constructs a contradiction, i.e. the limit is not necessarily greater than $\varepsilon$. So there exists some $a\in\widehat{A}$ such that for every $\varepsilon>0$, there exists integer $N>0$ such that $\|x_{v}-a\|_{2}<\varepsilon$ for all $v>N$.
Consequently, the group $\widehat{A}$ of equivalence classes of Cauchy sequences in $A$ is complete. We call it the completion of $A$.
Definition 6.3. Two seminorms $\|\cdot\|$ and $\|\cdot\|'$ on $A$ are equivalent if there exist $C,C'\in\mathbb{R}_{>0}$ such that $C\|f\|\leq\|f\|'\leq C'\|f\|$ for all $f\in M$.
Remark 6.4. Note that $C\|f\|\leq\|f\|'\leq C'\|f\|$ implies $\frac{1}{C'}\|f\|'\leq\|f\|\leq\frac{1}{C}\|f\|'$. And if there is a seminorm $\|\cdot\|''$ on $A$ such that $\|\cdot\|'$ and $\|\cdot\|''$ are equivalent, i.e. $C_{0}\|f\|'\leq\|f\|''\leq C_{0}'\|f\|'$ for some $C_{0},C_{0}'\in\mathbb{R}_{>0}$, then we have $CC_{0}\|f\|\leq\|f\|''\leq C'C_{0}'\|f\|$. These show that we have an equivalence relation.
Let $M$ be a seminormed group and $N$ be a subgroup of $M$. Then one can define the residue seminorm on $M/N$ by $\|f\|:=\inf\{\|g\|\mid g\in\pi^{-1}(f)\}$, where $\pi$ denotes the canonical homomorphism $M\rightarrow M/N$.
Lemma 6.5. Consider the situation above. The residue seminorm on $M/N$ is a norm if and only if $N$ is closed. If $M$ is a normed group whose norm is multiplicative and $N$ is closed, the residue norm on $M/N$ is multiplicative.
Proof. If $g\in f$, then $f-g\in N\Rightarrow g\in f+N$ for $f,g\in M$, we can write $g=f+n$ for some $n\in N$. Then $\|f\|=\inf_{g\in f}\|g\|=\inf_{n\in N}\|f+n\|=\inf_{n\in N}\|f-n\|=d(f,N)$. Since $\|f\|=0\Leftrightarrow d(f,N)=0\Leftrightarrow f\in\overline{N}$, $f=0$ if and only if $\overline{N}=N$.
If $f,g\in M/N$, then we have
$$ \begin{align*} \|fg\|=\inf\{\|g_{1}\|\mid g_{1}\in\pi^{-1}(fg)\}&\leq \inf\{\|g_{0}'\|\|g_{0}\|\mid g_{0}'g_{0}\in\pi^{-1}(f)\pi^{-1}(g)\} \\ &=\|f\|\|g\| \\ &=\inf\{\|g_{0}'g_{0}\|\mid g_{0}'g_{0}\in\pi^{-1}(fg)\} \\ &\leq \|fg\|, \end{align*} $$
which shows that $\|fg\|=\|f\|\|g\|$.
$\Box$
Definition 6.6. Let $\varphi: (M,\|\cdot\|)\rightarrow (N,\|\cdot\|')$ be a homomorphism of seminormed groups. We say that $\varphi$ is bounded if there exists $C>0$ such that $\|\varphi(f)\|'\leq C\|f\|$ for all $f\in M$. We say that $\varphi$ is admissible if identifying Im$(\varphi)$ with $M/\ker(\varphi)$ via the canonical isomorphism, the residue norm on $M/\ker(\varphi)$ is equivalent to the restriction to Im$(\varphi)$ of the seminorm of $N$.
Remark 6.7. Note that a homomorphism of seminormed groups is simply a homomorphism of groups which are seminormed.
Next, we consider the case of seminorms on rings. In the sequel, we let $A$ be a ring with unity.
Definition 6.8. A seminorm (resp. norm) on $A$ is a seminorm (resp. norm) on the additive group of $A$ such that $\|1\|=1$ and $\|ab\| \leq \|a\|\|b\|$ for all $a,b\in A$. A seminorm is power-multiplicative if $\|f^{n}\|=\|f\|^{n}$ for each integer $n>0$. A seminorm is multiplicative if it satisfies $\|ab\| = \|a\|\|b\|$ for $a,b\in A$.
Remark 6.9. Note that Definition 6.8 above is slightly different to that in [24], which says that a seminorm is multiplicative if it satisfies $\|ab\| = \|a\|\|b\|$ and $\|1\|=1$ for all $a,b\in A$.
Definition 6.10. A multiplicative norm is called a valuation, and a multiplicative seminorm is called a semivaluation.
Remark 6.11. Note that the notion of valuation in Definition 6.10 is different from what we have defined before (compared with the Definition 2.14, 2.20, and 2.21).
Definition 6.12. A ring endowed with a seminorm (resp. norm) is called seminormed (resp. normed) ring. A normed ring $\mathscr{A}$ is a Banach ring if $\mathscr{A}$ is complete with respect to its norm.
Definition 6.13. Let $\mathscr{A}$ be a normed ring. A seminormed (resp. normed) $\mathscr{A}$-module is an $\mathscr{A}$-module $M$ equipped with a seminorm (resp. norm) $\|\cdot\|$ such that $\|fm\|\leq C\|f\|\|m\|$ for some $C>0$ and for all $f\in\mathscr{A}$, $m\in M$. A Banach $\mathscr{A}$-module is a complete normed $\mathscr{A}$-module.
Definition 6.14. Let $(\mathscr{A},\|\cdot\|)$ be a Banach ring. A seminorm $\left|\cdot\right|$ on $\mathscr{A}$ is bounded if there exists $C>0$ such that $\left|f\right|\leq C\|f\|$ for all $f\in\mathscr{A}$.
Remark 6.15. In the previous definition, if $\left|\cdot\right|$ is a power-multiplicative bounded seminorm, then we can set $C=1$. In fact, by $\left|f^{n}\right|\leq C\|f^{n}\|$, we have $\left|f\right|\leq \sqrt[n]{C}\|f\|$ for all $n\geq1$, which implies that $\left|f\right|\leq\|f\|$.
Let $M,N$ be seminormed $\mathscr{A}$-module. We define the seminorm on $M\otimes_{\mathscr{A}}N$ by $\|f\|=\inf\sum_{i}\|m_{i}\|\cdot\|n_{i}\|$ where the infimum is taken over all representations $f=\sum_{i}m_{i}\otimes n_{i}$. We denote the completion of $M\otimes_{\mathscr{A}} N$ with respect to this seminorm by $M\widehat{\otimes}_{\mathscr{A}} N$. And we call $M\widehat{\otimes}_{\mathscr{A}} N$ the complete tensor product.
Let $\mathscr{A}$ be a commutative Banach ring with identity. The spectrum of $\mathscr{A}$ denoted by $\mathscr{M}(\mathscr{A})$ is the set of all bounded multiplicative seminorms on $\mathscr{A}$ provided with the weakest topology $\tau$ such that all real-valued functions on $\mathscr{M}(\mathscr{A})$ of the form $\varphi_{f}:\mathscr{M}(\mathscr{A})\rightarrow\mathbb{R}_{\geq0},\left|\cdot\right|\mapsto\left|f\right|$, $f\in\mathscr{A}$, are continuous, i.e. $\tau=\{\varphi^{-1}_{f}(U)\mid U\subset\mathbb{R}_{\geq0}\textrm{ open},f\in\mathscr{A}\}$.
Theorem 6.16. The spectrum $\mathscr{M}(\mathscr{A})$ is a non-empty and compact Hausdorff space.
Proof. cf. [25, Theorem 1.2.1, p13]
$\Box$
Definition 6.17. Let $k$ be a commutative Banach ring. The $n$-dimensional affine space $A^{n}=A^{n}_{k}$ over $k$ is the set of all multiplicative seminorms on $k[T_{1}, T_{2}, ..., T_{n}]$ whose restriction to $k$ is bounded. The set is provided with the weakest topology such that all real-valued functions on $A^{n}$ of the form $\left|\cdot\right|\mapsto\left|f\right|$, $f\in k[T_{1}, T_{2}, ..., T_{n}]$, are continuous.
We fix a non-archimedean field $k$ with a valuation (multiplicative norm) $\left|\cdot\right|$ which may be trivial. We set $\sqrt{\left|k^{*}\right|}:=\{\alpha\in\mathbb{R}_{\geq0}\mid\alpha^{n}\in\left|k^{*}\right| \textrm{for some }n\geq1\}$.
For $r_{1},r_{2},...,r_{n}>0$, we set:
$$ k\{r_{1}^{-1}T_{1}, ..., r_{n}^{-1}T_{n}\}\overset{\textrm{def}}=\{f=\sum_{v=0}^{\infty}a_{v}T^{v}\mid a_{v}\in k\textrm{ and }\left|a_{v}\right|r^{v}\to0\textrm{ as }\left|v\right|\to\infty\} $$
here $v=(v_{1},v_{2},...,v_{n})$, $\left|v\right|=v_{1}+v_{2}+\cdot\cdot\cdot+v_{n}$, $T^{v}=T^{v_{1}}_{1}T^{v_{2}}_{2}\cdot\cdot\cdot T^{v_{n}}_{n}$ and $r^{v}=r^{v_{1}}_{1}r^{v_{2}}_{2}\cdot\cdot\cdot r^{v_{n}}_{n}$. This is a commutative Banach $k$-algebra with respect to the multiplicative norm $\|f\|=\max_{v}\left|a_{v}\right|r^{v}$. And this algebra will be simply denoted by $k\{r^{-1}T\}$.
The construction above can be extended to $\mathscr{A}\{r^{-1}T\}$ for any commutative Banach $k$-algebra $\mathscr{A}$.
Example 6.18. Suppose that the valuation on $k$ is trivial. If $r_{i}\geq1$ for all $1\leq i\leq n$, then $k\{r^{-1}T\}$ becomes the ring of polynomials $k[T_{1},T_{2},...,T_{n}]$. If $r_{i}<1$ for all $1\leq i\leq n$, then $k\{r^{-1}T\}$ becomes the ring of formal power series $k[[T_{1},T_{2},...,T_{n}]]$.
Definition 6.19. A commutative Banach $k$-algebra $\mathscr{A}$ is said to be $k$-affinoid if there exists an admissible epimorphism $k\{r^{-1}T\}\rightarrow\mathscr{A}$. If such an epimorphism can be found with $r=1$, $\mathscr{A}$ is said to be strictly $k$-affinoid. An affinoid $k$-algebra is a $K$-affinoid algebra for some non-archimedean field $K$ over $k$.
Definition 6.20. A $k$-affinoid variety is a pair $\textrm{Sp }A=(\textrm{Max }A, A)$, where $A$ is a $k$-affinoid algebra and Max $A$ is the set of all maximal ideals of $A$. A morphism $\varphi: \textrm{Sp }A\rightarrow \textrm{Sp }B$ of affinoid varieties (also called a $k$-affinoid morphism or a $k$-affinoid map) is a pair $(^{a}\sigma,\sigma)$, where $\sigma: B\rightarrow A$ is a $k$-algebra homomorphism and $^{a}\sigma: \textrm{Max }A\rightarrow \textrm{Max }B$ is a map induced from $\sigma$.
Remark 6.21. We will use Sp $A$ in the sense of Max $A$ in the sequel. For any affinoid variety Sp $A$, the elements in the affinoid algebra $A$ are called affinoid functions on Sp $A$.
Definition 6.22. Let $\mathscr{A}$ be $k$-affinoid algebra. A Banach $\mathscr{A}$-module $M$ is finite if there exists an admissible epimorphism $\mathscr{A}^{n}\rightarrow M$. A commutative Banach $\mathscr{A}$-algebra $\mathscr{B}$ is finite if $\mathscr{B}$ is finite as a Banach $\mathscr{A}$-module.
Proposition 6.23. If the valuation on $k$ is non-trivial, then for a strictly $k$-affinoid algebra $\mathscr{A}$ the canonical map ${\rm{Max}}(\mathscr{A})\rightarrow\mathscr{M}(\mathscr{A})$ induces a homeomorphism of ${\rm{Max}}(\mathscr{A})$ with an everywhere dense subset of $\mathscr{M}(\mathscr{A})$.
Proof. cf. [25, Proposition 2.1.15, p26].
$\Box$
Remark 6.24. Everywhere dense is the same as dense.