Let $X=\mathbb{P}_{K}^{n}$ and $L=\mathcal{O}(1)$, the universal line bundle of ${\mathbb{P}_{K}^{n}}$. As a model for $(X,L)$ we consider $\mathcal{X}=\mathbb{P}_{K^{\circ}}^{n}$, the projective space over $\operatorname{Spec}(K^{\circ})$, $\mathcal{L}=\mathcal{O}_{\mathbb{P}_{K^{\circ}}^{n}}(1)$, and $e=1$. A rational section $s$ of $L$ can be identified with a homogeneous rational function $\rho_{s}\in K(x_{0},\dots,x_{n})$ of degree 1.

Let $p=(p_{0}:\dots:p_{n})\in(\mathbb{P}_{K}^{n})^{\text{\rm an}}\setminus\operatorname{div}(s)$ and set $H=\mathscr{H}(p)$. Let $i_{0}$ be such that $|p_{i_{0}}|=\max_{i}\{|p_{i}|\}$. Take $U\simeq\mathbb{A}_{K}^{n}$ (respectively $\mathcal{U}\simeq\mathbb{A}_{K^{\circ}}^{n}$) as the affine set $x_{i_{0}}\not=0$ over $H$ (respectively $H^{\circ}$). The point $p$ corresponds to the algebraic morphism

$$p^{\ast}\colon K[X_{0},\dots,X_{i_{0}-1},X_{i_{0}+1},\dots,X_{n}]\longrightarrow H$$

that sends $X_{i}$ to $p_{i}/p_{i_{0}}$. The extension ${\widetilde{p}}$ factors through the algebraic morphism

$${\widetilde{p}}^{\ast}\colon K^{\circ}[X_{1},\dots,X_{i_{0}-1},X_{i_{0}+1},\dots,X_{n}]\longrightarrow H^{\circ},$$

with the same definition. Then

	$\displaystyle||s(p)||$	$\displaystyle=\inf\big\{|z|\ \big|z\in H^{\times},z^{-1}{\widetilde{p}}^{\ast}s\in{\widetilde{p}}^{*}{\mathcal{L}}\big\}$
		$\displaystyle=\inf\big\{|z|\ \big|z\in H^{\times},z^{-1}\rho_{s}(p_{0}/p_{i_{0}},\dots,1,\dots,p_{n}/p_{i_{0}})\in H^{\circ}\big\}$
		$\displaystyle=\left|\frac{\rho_{r}(p_{0},\dots,p_{n})}{p_{i_{0}}}\right|$
		$\displaystyle=\frac{|\rho_{r}(p_{0},\dots,p_{n})|}{\max_{i}\{|p_{i}|\}}.$

We call this the canonical metric of ${\mathcal{O}}(1)^{{\text{\rm an}}}$ and we denote it by $\|\cdot\|_{{\operatorname{can}}}$.