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

$$\forall\,s\in H^{0}(X,L),\quad\|s\|_{h}:=\sup_{x\in X^{\mathrm{an}}}|s|_{h}(x).$$

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

$$\forall\,l\in H^{0}(Y,L),\quad\|l\|_{Y,h}:=\sup_{y\in Y^{\mathrm{an}}}|l|_{h}(y).$$

Clearly one has

(1)

$$\|s\|_{h}\geqslant\|{\left.{s}\right|_{{Y}}}\|_{Y,h}$$

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