We assume that there is a subspace $H$ of $H^{0}(X,L)$ such that $H\otimes_{k}\mathscr{O}_{X}\to L$ is surjective and the morphism $\phi_{H}:X\to\mathbb{P}(H)$ induced by $H$ is a closed embedding. We identify $X$ with $\phi_{H}(X)$, so that $L=\left.{\mathscr{O}_{\mathbb{P}(H)}(1)}\right|_{{X}}$. Let $\|\raisebox{1.72218pt}{.}\|$ be a norm of $H$ such that $H$ has an orthonormal basis $(e_{1},\ldots,e_{r})$ with respect to $\|\raisebox{1.72218pt}{.}\|$. We set

$$h:=\left\{|\raisebox{1.72218pt}{.}|^{\mathrm{quot}}_{(H,\|\raisebox{1.20552pt}{.}\|)}(x)\right\}_{x\in X^{\mathrm{an}}}\quad\text{and}\quad\mathscr{H}:=\mathfrak{o}_{k}e_{1}+\cdots+\mathfrak{o}_{k}e_{r}=(H,\|\raisebox{1.72218pt}{.}\|)_{\leq 1}.$$

Let $\mathscr{X}$ be the Zariski closure of $X$ in $\mathbb{P}(\mathscr{H})$ (cf. §1.1.7) and $\mathscr{L}:=\left.{\mathscr{O}_{\mathbb{P}(\mathscr{H})}(1)}\right|_{{\mathscr{X}}}$. Then $|\raisebox{1.72218pt}{.}|_{h}(x)=|\raisebox{1.72218pt}{.}|_{\mathscr{L}}(x)$ for all $x\in X^{\mathrm{an}}$.