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

$$|\omega^{\otimes n}|_{h^{n}}(x)=|\omega|_{h}(x)^{n}.$$

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 $M^{\otimes n}$.