Fix a field $k$ with a complete and non-archimedean absolute value $|\raisebox{1.72218pt}{.}|$. The valuation ring of $k$ and the maximal ideal of the valuation ring are denoted by $\mathfrak{o}_{k}$ and $\mathfrak{m}_{k}$, respectively, that is,

In the case where $|\raisebox{1.72218pt}{.}|$ is discrete, we fix a uniformizing parameter $\varpi$ of $\mathfrak{m}_{k}$, that is, $\mathfrak{m}_{k}=\varpi\mathfrak{o}_{k}$.

A norm $\|\raisebox{1.72218pt}{.}\|$ of a finite-dimensional vector space $V$ over $k$ is always assumed to be ultrametric, that is, $\|x+y\|\leq\max\{\|x\|,\|y\|\}$. A pair $(V,\|\raisebox{1.72218pt}{.}\|)$ is called a normed finite-dimensional vector space over $k$.

Fix an algebraic scheme $X$ over $\operatorname{Spec}k$, that is, $X$ is a scheme of finite type over $\operatorname{Spec}(k)$. Let $X^{\mathrm{an}}$ be the analytification of $X$ in the sense of Berkovich [1]. For $x\in X^{\mathrm{an}}$, the residue field of the associated scheme point of $x$ is denoted by $\kappa(x)$. Note that the seminorm $|\raisebox{1.72218pt}{.}|_{x}$ at $x$ yields an absolute value of $\kappa(x)$. By abuse of notation, it is denoted by $|\raisebox{1.72218pt}{.}|_{x}$. Let $\hat{\kappa}(x)$ be the completion of $\kappa(x)$ with respect to $|\raisebox{1.72218pt}{.}|_{x}$. The extension of $|\raisebox{1.72218pt}{.}|_{x}$ to $\hat{\kappa}(x)$ is also denoted by the same symbol $|\raisebox{1.72218pt}{.}|_{x}$. The valuation ring of $\hat{\kappa}(x)$ and the maximal ideal of the valuation ring are denoted by $\mathfrak{o}_{x}$ and $\mathfrak{m}_{x}$, respectively. Let $L$ be an invertible sheaf on $X$. For $x\in X^{\operatorname{an}}$, $L\otimes_{{\mathscr{O}}_{X}}\hat{\kappa}(x)$ is denoted by $L(x)$.

By continuous metric on $L$, we refer to a family $h=\{|\raisebox{1.72218pt}{.}|_{h}(x)\}_{x\in X^{\mathrm{an}}}$, where $|\raisebox{1.72218pt}{.}|_{h}(x)$ is a norm on $L\otimes_{\mathscr{O}_{X}}\hat{\kappa}(x)$ over $\hat{\kappa}(x)$ for each $x\in X^{\mathrm{an}}$, such that for any local basis $\omega$ of $L$ over a Zariski open subset $U$, $|\omega|_{h}(\raisebox{1.72218pt}{.})$ is a continuous function on $U^{\mathrm{an}}$. We assume that $X$ is projective. Given a continuous metric $h$ on $L$, we define a norm $\|\raisebox{1.72218pt}{.}\|_{h}$ on $H^{0}(X,L)$ such that

Similarly, if $Y$ is a closed subscheme of $X$, we define a norm $\|\raisebox{1.72218pt}{.}\|_{Y,h}$ on $H^{0}(Y,L)$ such that

Clearly one has

for any $s\in H^{0}(X,L)$.

Given a continuous metric $h$ on $L$, the metric induces for each integer $n\geqslant 1$ a continuous metric on $L^{\otimes n}$ which we denote by $h^{n}$: for any point $x\in X^{\mathrm{an}}$ and any local basis $\omega$ of $L$ over a Zariski open neighborhood of $x$ one has

Note that for any section $s\in H^{0}(X,L)$ one has $\|s^{\otimes n}\|_{h^{n}}=\|s\|_{h}^{n}$. By convention, $h^{0}$ denotes the trivial metric on $L^{\otimes 0}=\mathscr{O}_{X}$, namely $|\mathbf{1}|_{h^{0}}(x)=1$ for any $x\in X^{\mathrm{an}}$, where $\mathbf{1}$ denotes the section of unity of $\mathscr{O}_{X}$.

Conversely, given a continuous metric $g=\{|\raisebox{1.72218pt}{.}|_{g}(x)\}_{x\in X^{\mathrm{an}}}$ on $L^{\otimes n}$, there is a unique continuous metric $h$ on $L$ such that $h^{n}=g$. We denote by $g^{1/n}$ this metric. This observation allows to define continuous metrics on an element in $\mathrm{Pic}(X)\otimes\mathbb{Q}$ as follows. Given $M\in\operatorname{Pic}(X)\otimes\mathbb{Q}$, we denote by $\Gamma(M)$ the subsemigroup of $\mathbb{N}_{\geq 1}$ of all positive integers $n$ such that $M^{\otimes n}\in\operatorname{Pic}(X)$. We call continuous metric on $M$ any family $g=(g_{n})_{n\in\Gamma(M)}$ with $g_{n}$ being a continuous metric on $M^{\otimes n}$, such that $g_{n}^{m}=g_{mn}$ for any $n\in\Gamma(M)$ and any $m\in\mathbb{N}_{\geq 1}$. Note that the family $g=(g_{n})_{n\in\Gamma(M)}$ is uniquely determined by any of its elements. In fact, given an element $n\in\Gamma(M)$, one has $g_{m}=g_{mn}^{1/n}=(g_{n}^{m})^{1/n}$ for any $m\in\Gamma(M)$. In particular, for any positive rational number $p/q$, the family $g^{p/q}=(g_{Nnp}^{1/Nq})_{n\in\Gamma(M^{\otimes(p/q)})}$ is a continuous metric on $M^{\otimes(p/q)}$, where $N$ is a positive integer such that $M^{\otimes N}\in\operatorname{Pic}(X)$, and the metric $g^{p/q}$ does not depend on the choice of the positive integer $N$.

Let $M$ be an element in $\operatorname{Pic}(X)\otimes\mathbb{Q}$ equipped with a continuous metric $g=(g_{n})_{n\in\Gamma(M)}$. By abuse of notation, for $n\in\Gamma(M)$ we also use the expression $g^{n}$ to denote the continuous metric $g_{n}$ on $M^{\otimes n}$.

We call model of $X$ any projective and flat $\mathfrak{o}_{k}$-scheme $\mathscr{X}\to\operatorname{Spec}(\mathfrak{o}_{k})$ such that the generic fiber of $\mathscr{X}\to\operatorname{Spec}(\mathfrak{o}_{k})$ is $X$. We denote by $\mathscr{X}_{\circ}:=\mathscr{X}\otimes_{\mathfrak{o}_{k}}(\mathfrak{o}_{k}/\mathfrak{m}_{k})$ the central fiber of $\mathscr{X}\to\operatorname{Spec}(\mathfrak{o}_{k})$. By the valuative criterion of properness, for any point $x\in X^{\mathrm{an}}$, the canonical $k$-morphism $\operatorname{Spec}\hat{\kappa}(x)\rightarrow X$ extends in a unique way to an $\mathfrak{o}_{k}$-morphism of schemes $\mathscr{P}_{x}:\operatorname{Spec}\mathfrak{o}_{x}\rightarrow\mathscr{X}$. We denote by $r_{\mathscr{X}}(x)$ the image of $\mathfrak{m}_{x}\in\operatorname{Spec}\mathfrak{o}_{x}$ by the map $\mathscr{P}_{x}$. Thus we obtain a map $r_{\mathscr{X}}$ from $X^{\mathrm{an}}$ to $\mathscr{X}_{\circ}$, called the reduction map of $\mathscr{X}$.

Let $\mathscr{L}$ be an element of $\operatorname{Pic}(\mathscr{X})\otimes{\mathbb{Q}}$ such that $\left.{\mathscr{L}}\right|_{{X}}=L$ in $\operatorname{Pic}(X)\otimes{\mathbb{Q}}$. The ${\mathbb{Q}}$-invertible sheaf $\mathscr{L}$ yields a continuous metric $|\raisebox{1.72218pt}{.}|_{\mathscr{L}}$ as follows.

First we assume that $\mathscr{L}\in\operatorname{Pic}(\mathscr{X})$ and $\left.{\mathscr{L}}\right|_{{X}}=L$ in $\operatorname{Pic}(X)$. For any $x\in X^{\mathrm{an}}$, let $\omega_{x}$ be a local basis of $\mathscr{L}$ around $r_{\mathscr{X}}(x)$ and $\bar{\omega}_{x}$ the class of $\omega_{x}$ in ${L(x)}:=L\otimes_{\mathscr{O}_{X}}\hat{\kappa}(x)$. For $l\in L\otimes_{\mathscr{O}_{X}}\hat{\kappa}(x)$, if we set $l=a_{x}\bar{\omega}_{x}$ ($a_{x}\in\hat{\kappa}(x)$), then $|l|_{\mathscr{L}}(x):=|a_{x}|_{x}$. Here we set $h:=\{|\raisebox{1.72218pt}{.}|_{\mathscr{L}}(x)\}_{x\in X^{\mathrm{an}}}$. Note that $h$ is continuous because, for a local basis $\omega$ of $\mathscr{L}$ over an open set $\mathscr{U}$ of $\mathscr{X}$, $|\omega|_{\mathscr{L}}(x)=1$ for all $x\in r_{\mathscr{X}}^{-1}(\mathscr{U}_{\circ})$. Moreover,

for all $n\geq 0$ and $x\in X^{\mathrm{an}}$. Indeed, if we set $l=a_{x}\bar{\omega}_{x}$ for $l\in{L(x)}$, then $l^{\otimes n}=a_{x}^{n}\bar{\omega}_{x}^{\otimes n}$. Thus

In general, there are $\mathscr{M}\in\operatorname{Pic}(\mathscr{X})$ and a positive integer $m$ such that $\mathscr{L}^{\otimes m}=\mathscr{M}$ in $\operatorname{Pic}(\mathscr{X})\otimes{\mathbb{Q}}$ and $\left.{\mathscr{M}}\right|_{{X}}=L^{\otimes m}$ in $\operatorname{Pic}(X)$. Then

Note that the above definition does not depend on the choice of $\mathscr{M}$ and $m$. Indeed, let $\mathscr{M}^{\prime}$ and $m^{\prime}$ be another choice. As $\mathscr{M}^{\otimes m^{\prime}}=\mathscr{M^{\prime}}^{\otimes m}$ in $\operatorname{Pic}(\mathscr{X})\otimes{\mathbb{Q}}$, there is a positive integer $N$ such that $\mathscr{M}^{\otimes Nm^{\prime}}=\mathscr{M^{\prime}}^{\otimes Nm}$ in $\operatorname{Pic}({\mathscr{X}})$, so that, by using (2),

as desired.

Let $\mathscr{X}$ be a model of $X$. As $\mathscr{X}$ is flat over $\mathfrak{o}_{k}$, the natural homomorphism $\mathscr{O}_{\mathscr{X}}\to\mathscr{O}_{X}$ is injective. Let $Y$ be a closed subscheme of $X$ and $I_{Y}\subseteq\mathscr{O}_{X}$ the defining ideal sheaf of $Y$. Let $\mathscr{I}_{\mathscr{Y}}$ be the kernel of $\mathscr{O}_{\mathscr{X}}\to\mathscr{O}_{X}/I_{Y}$, that is, $\mathscr{I}_{\mathscr{Y}}:=\mathscr{I}_{Y}\cap\mathscr{O}_{\mathscr{X}}$. Obviously $\mathscr{I}_{\mathscr{Y}}\otimes_{\mathfrak{o}_{k}}k=I_{Y}$, so that if we set $\mathscr{Y}=\operatorname{Spec}(\mathscr{O}_{\mathscr{X}}/\mathscr{I}_{\mathscr{Y}})$, then $\mathscr{Y}\times_{\operatorname{Spec}(\mathfrak{o}_{k})}\operatorname{Spec}(k)=Y$. Moreover, $\mathscr{Y}$ is flat over $\mathfrak{o}_{k}$ because $\mathscr{O}_{\mathscr{Y}}\to\mathscr{O}_{Y}$ is injective. Therefore, $\mathscr{Y}$ is a model of $Y$. We say that $\mathscr{Y}$ is the Zariski closure of $Y$ in $\mathscr{X}$.

For $\alpha\in(0,1]$, a basis $(e_{1},\ldots,e_{r})$ of $V$ is called an $\alpha$-orthogonal basis of $V$ with respect to $\|\raisebox{1.72218pt}{.}\|$ if

If $\alpha=1$ (resp. $\alpha=1$ and $\|e_{1}\|=\cdots=\|e_{r}\|=1$), then the above basis is called an orthogonal basis of $V$ (resp. an orthonormal basis of $V$). Let $(e^{\prime}_{1},\ldots,e^{\prime}_{r})$ be another basis of $V$. We say that $(e_{1},\ldots,e_{r})$ is compatible with $(e^{\prime}_{1},\ldots,e^{\prime}_{r})$ if $ke_{1}+\cdots+ke_{i}=ke^{\prime}_{1}+\cdots+ke^{\prime}_{i}$ for $i=1,\ldots,r$.

Let $V^{\prime}$ be a vector space over $k$ and $\|\raisebox{1.72218pt}{.}\|^{\prime}$ a norm of $V^{\prime}$.

By the above lemma, we define $\|\phi\|_{\mathrm{Hom}_{k}(V,V^{\prime})}$ to be

Note that $\|\raisebox{1.72218pt}{.}\|_{\mathrm{Hom}_{k}(V,V^{\prime})}$ yields a norm on $\mathrm{Hom}_{k}(V,V^{\prime})$. We denote $\|\raisebox{1.72218pt}{.}\|_{\mathrm{Hom}_{k}(V,k)}$ by $\|\raisebox{1.72218pt}{.}\|^{\vee}$ (i.e. the case where $V^{\prime}=k$ and $\|\raisebox{1.72218pt}{.}\|^{\prime}=|\raisebox{1.72218pt}{.}|$).