Let $X$ be a strictly $K$-analytic space and $L$ a line bundle on $X$, i.e. a locally free sheaf of rank 1 on the G-topology. A continuous metric $\|\cdot\|$ on $L$ is a function which asserts to any admissible open subset $U\subseteq X$ and any section $s\in\Gamma(U,L)$ a continuous (with respect to the Berkovich topology) function $\|s(\cdot)\|:U\rightarrow\mathbb{R}_{\geq 0}$ such that:

1. i)

   For an admissible open subset $V\subseteq U$ we have $\left\|s\Big|_{V}(\cdot)\right\|=\|s(\cdot)\|\Big|_{V}$,

2. ii)

   for $f\in\Gamma(U,\mathcal{O}_{X})$ we have $\|fs(\cdot)\|=|f(\cdot)|\|s(\cdot)\|$,

3. iii)

   for $p\in U$ we have $\|s(p)\|=0$ if and only if $s(p)=0$.

Given a formal model $(\mathfrak{X},\mathfrak{L})$ of $(X,L)$ one can define an associated so called formal metric $\|\cdot\|_{\mathfrak{L}}$ on $L$ in the following way: If $s$ is a local frame of $\mathfrak{L}$ on a formal open subset $\mathfrak{U}\subseteq\mathfrak{X}$ we define $\|fs(\cdot)\|_{\mathfrak{L}}=|f(\cdot)|$ on $\mathfrak{U}^{\textup{an}}$ for any $f\in\Gamma(\mathfrak{U}^{\textup{an}},\mathcal{O}_{\mathfrak{X}}^{\textup{an}})$. As this is independent of the choice of $s$ and $\mathfrak{X}^{\textup{an}}$ is covered by such sets, this gives a well-defined metric on $L$.

Let $X$ be a proper scheme over $K$ and $L$ a line bundle on $X$. An algebraic $K^{\circ}$-model of $X$ is a proper flat scheme $\mathscr{X}$ over $K^{\circ}$ with a fixed isomorphism from the generic fibre $\mathscr{X}_{\eta}$ to $X$. An algebraic $K^{\circ}$-model of $(X,L)$ is a pair $(\mathscr{X},\mathscr{L})$ where $\mathscr{X}$ is an algebraic $K^{\circ}$-model of $X$ and $\mathscr{L}$ is a line bundle on $\mathscr{X}$ with a fixed isomorphism from $\mathscr{L}\Big|_{X}$ to $L$. An algebraic $K^{\circ}$-model of $(X,L)$ gives rise to a formal $K^{\circ}$-model of $(X^{\textup{an}},L^{\textup{an}})$ by formal completion. Hence by the above, an algebraic model of $(X,L)$ induces a formal metric on $L^{\textup{an}}$. We call such metrics algebraic metrics.

Let $X$ be a strictly $K$-analytic space and $L$ a line bundle on $X$. A metric $\|\cdot\|$ on $L$ is called piecewise linear if there is a G-covering $(V_{i})_{i\in I}$ and frames $s_{i}$ of $L$ over $V_{i}$ for every $i\in I$ such that $\|s_{i}(\cdot)\|=1$ on $V_{i}$.

Let $X$ be a strictly $K$-analytic space and $L$ a line bundle on $X$. A piecewise linear metric on $L$ is called semipositive in $x\in X$ if there exists a compact strictly $K$-analytic domain $W$ which is a neighbourhood of $x$ such that there is a formal model $(\mathfrak{W},\mathfrak{L})$ of $\left(W,L\Big|_{W}\right)$ inducing the metric on $W$ and satisfying $\Deg_{\mathfrak{L}}(C)\geq 0$ for every proper closed curve $C$ in the special fibre of $\mathfrak{W}$. The metric on $L$ is called semipositive in a subset $V\subseteq X$ if it is semipositive in every $x\in V$. It is called semipositive if it is semipositive in $X$.

Let $X$ be a strictly $K$-analytic space and $L$ a line bundle on $X$. A metric $\|\cdot\|$ on $L$ is called piecewise $\mathbb{Q}$-linear if for every $x\in X$ there is an open neighbourhood $W$ of $x$ and a non-zero $n\in\mathbb{N}$ such that $\|\cdot\|^{\otimes n}\Big|_{W}$ is a piecewise linear metric on $L^{\otimes n}\Big|_{W}$.
A piecewise $\mathbb{Q}$-linear metric on $L$ is called semipositive in $x\in X$ if in the above $\|\cdot\|^{\otimes n}\Big|_{W}$ is semipositive in $x$.