A metric on $L^{{\text{\rm an}}}$ is an assignment that, to each local section $s$ of $L^{{\text{\rm an}}}$ on an open subset $U\subset X^{{\text{\rm an}}}$, associates a continuous function

$$\|s(\cdot)\|\colon U\longrightarrow\mathbb{R}_{\geq 0}$$

such that, for all $p\in U$,

1. (1)

   $\|s(p)\|=0$ if and only if $s(p)=0$;

2. (2)

   for any $\lambda\in\mathcal{O}_{X^{{\text{\rm an}}}}(U)$, it holds $\|(\lambda s)(p)\|=|\lambda(p)|\,\|s(p)\|.$

The pair ${\overline{L}}:=(L,\|\cdot\|)$ is called a metrized line bundle.The metric $\|\cdot\|$ is smooth if for every local section $s$ of $L^{{\text{\rm an}}}$, the function $\|s(\cdot)\|^{2}$ is smooth.