Since this is stated here under more general assumptions than in [GK15, Proposition 5.11], we sketch the argument. Let $({\mathfrak{W}},{\mathfrak{L}})$ be the ${K^{\circ}}$-model for the given formal metric on $L|_{W}$. We may assume that ${\mathfrak{W}}$ is a formal open subset of a formal ${K^{\circ}}$-model ${\mathfrak{V}}$ of $V$ [Bos14, Lemma 8.4.5]. By the argument in [BL93a, Lemma 5.7], there is a coherent ${\mathcal{O}}_{\mathfrak{V}}$-module ${\mathscr{F}}$ on ${\mathfrak{V}}$ which extends ${\mathfrak{L}}$. This works even for paracompact $V$ as noted in the proof of [CD12, Proposition 6.2.13] and the argument there (or in the proof of [Gub98, Lemma 7.6]) shows that after replacing ${\mathfrak{V}}$ by a suitable admissible blowing-up, we may assume that ${\mathscr{F}}$ is a line bundle. Then the associated formal metric satisfies the claim. ∎

Clearly, every formal metric is piecewise linear. To prove the converse, we may assume that $V$ is connected. It is a general fact from topology (see [Bou71, chap. 1, §9, Théorème 5]) that a connected locally compact space is paracompact if and only if it is countable at infinity. It follows that there is a finite or a countable ${\rm G}$-open covering $(V_{i})_{i\in I}$ of $V$ of finite type by strictly $K$-affinoid domains $V_{i}$ with frames $s_{i}$ of $L|_{V_{i}}$ such that $\|s_{i}\|=1$ on $V_{i}$. Then $V_{i}$ is the Berkovich spectrum of a strictly $K$-affinoid algebra ${\mathscr{A}}_{i}$. Obviously, there is an admissible ${K^{\circ}}$-algebra $A_{i}$ with ${\mathscr{A}}_{i}=A_{i}\otimes_{K^{\circ}}K$. For $f_{i}\in{\mathscr{A}}_{i}^{\circ}$, the ${K^{\circ}}$-algebra $A_{i}[f_{i}]$ is

an admissible ${K^{\circ}}$-algebra [Bos14, Lemma 8.4.6].

Using the existence of a formal metric on $L^{\rm an}$, we may assume that $L={\mathcal{O}}_{V}$ and hence the frames $s_{i}$ are invertible functions on the sets $V_{i}$. Using that $V$ is paracompact, the underlying rigid space is quasiseparated and hence $V_{ij}=V_{i}\cap V_{j}={\rm Spf}({\mathscr{A}}_{ij})$ for some strictly $K$-affinoid algebra ${\mathscr{A}}_{ij}$. If $f_{ij}=s_{i}/s_{j}$, then $f_{ij}\in({\mathscr{A}}_{ij}^{\circ})^{\times}$. Using the above, we choose a formal affine ${K^{\circ}}$-model ${\mathfrak{W}}_{ij}$ with generic fiber $V_{ij}$ such that $f_{ij}\in{\mathcal{O}}({\mathfrak{W}}_{ij})$.

In the following, we assume that $I={\mathbb{N}}\setminus\{0\}$ (the finite case is similar and even easier) and we consider $k\in{\mathbb{N}}$. By an inductive procedure, we will construct a formal model ${\mathfrak{V}}^{(k)}$ of $V$ such that $V_{i}$ is the generic fiber of a formal open subset ${\mathfrak{V}}_{i}^{(k)}$ of ${\mathfrak{V}}^{(k)}$ for every $i\in I$ and such that ${\mathfrak{V}}_{i}^{(k)}\cap{\mathfrak{V}}_{j}^{(k)}$ is lying over ${\mathfrak{W}}_{ij}$ for every $i,j\in\{0,\dots,k\}$. By this we mean that for every $i,j\in\{0,\dots,k\}$ there exists a morphism ${\mathfrak{V}}_{i}^{(k)}\cap{\mathfrak{V}}_{j}^{(k)}\to{\mathfrak{W}}_{ij}$ which is the identity on the generic fibre.

Note that the case $k=0$ follows from [Bos14, Lemma 8.4.5]. Let $k\geq 1$ and assume that ${\mathfrak{V}}^{(k-1)}$ is already constructed. By Raynaud’s theorem and [BL93b, Corollary 5.4], there is an admissible formal blowing up $p_{k}$ of ${\mathfrak{V}}_{k}^{(k-1)}$ such that $V_{ik}$ (resp. $V_{ki}$) is the generic fiber of a formal open subset lying over ${\mathfrak{W}}_{ik}$ (resp. ${\mathfrak{W}}_{ki}$) for $i=1,\dots,k$. By [Bos14, Proposition 8.2.13], we may extend $p_{k}$ to an admissible formal blowing up ${\mathfrak{V}}^{(k)}$ of ${\mathfrak{V}}^{(k-1)}$ with center $Z$ in the special fiber such that $Z$ is disjoint from every ${\mathfrak{V}}_{i}^{(k-1)}$ with $i\leq k-1$ satisfying ${\mathfrak{V}}_{i}^{(k-1)}\cap{\mathfrak{V}}_{k}^{(k-1)}=\emptyset$. Then ${\mathfrak{V}}^{(k)}$ satisfies the claim with ${\mathfrak{V}}_{i}^{(k)}$ equal to the preimage of ${\mathfrak{V}}_{i}^{(k-1)}$ in ${\mathfrak{V}}^{(k)}$.

Using that the $\rm G$-covering $(V_{i})_{i\in I}$ is of finite type, the above construction shows that the formal models ${\mathfrak{V}}^{(k)}$ eventually become stable over ${\mathfrak{V}}_{i}^{(0)}$ for any $i\in I$ and hence we get a formal model ${\mathfrak{V}}$ of $V$ lying above all the models ${\mathfrak{V}}^{(k)}$. It has the property that every $V_{i}$ is the generic fiber of a formal open subset ${\mathfrak{V}}_{i}$ and that ${\mathfrak{V}}_{i}\cap{\mathfrak{V}}_{j}$ is lying over ${\mathfrak{W}}_{ij}$ for every $i,j\in I$. Since $f_{ij}$ and $f_{ji}$ are both in ${\mathcal{O}}({\mathfrak{V}}_{i}\cap{\mathfrak{V}}_{j})$, we see that $f_{ij}$ is invertible on ${\mathfrak{V}}_{i}\cap{\mathfrak{V}}_{j}$. This means that $(s_{i})_{i\in I}$ is a vertical Cartier divisor on ${\mathfrak{V}}$ inducing the metric. ∎

These properties are proved in [Gub98, Section 7] under the assumption that $K$ is algebraically closed and $V$ is compact. The assumption $K$ algebraically closed was not used in the arguments. Since (a)–(d) are local statements, we can deduce them from the corresponding statements in loc.  cit. ∎

By compactness of $U\setminus U^{\circ}$, there exists a compact strictly $K$-analytic domain $Z\subset V$ such that $Z$ is a neighbourhood of $U\setminus U^{\circ}$ and $W\cap Z=\emptyset$. Hence $W\coprod Z$ is a compact strictly $K$-analytic domain of $V$ and we consider the piecewise linear function on $W\coprod Z$ defined by $f$ on $W$ and by $0$ on $Z$. Then we apply Proposition 2.6 to $L=\mathcal{O}_{V}$, in which case formal metrics correspond to piecewise linear functions (see Proposition 2.8). We deduce that there exists a piecewise linear function $g\colon V\to{\mathbb{R}}$ which agrees with $f$ on $W$ and which agrees with $0$ on $Z$. But since $Z$ is a neighborhood of $U\setminus U^{\circ}$, we deduce that the function $\varphi\colon V\to{\mathbb{R}}$ defined by

$$\varphi(x)=\begin{cases}g(x)&{\rm if}\ x\in U\\ 0&{\rm if}\ x\notin U\end{cases}$$

is still piecewise linear. Since $\varphi$ extends $f$ and ${\rm supp}(\varphi)\subset U$, we get the claim. ∎

Since piecewise ${\mathbb{Q}}$-linear functions are dense in the compact case [Gub98, Theorem 7.12], there exists a piecewise ${\mathbb{Q}}$-linear function $g\colon W\to{\mathbb{R}}$ such that $f-\varepsilon\leq g\leq f$ on $W$. Since $W$ is compact, there is a non-zero $k\in{\mathbb{N}}$ such that $kg$ is piecewise linear. By Proposition 2.6 and Proposition 2.8 applied to the formal metric on ${\mathcal{O}}_{V}$ associated to $kg$, there exists a piecewise ${\mathbb{Q}}$-linear function $\psi\colon V\to{\mathbb{R}}$ which extends $g$. We then set $\varphi\coloneqq\max(\psi,0)$. By Proposition 2.10 (d), $\varphi$ is piecewise ${\mathbb{Q}}$-linear. By definition, we have $\varphi\geq 0$. We have $\psi\leq f$ on $W$ and $f$ is non-negative, hence we have $\varphi\leq f$ on $W$. Finally, since $f-\varepsilon\leq\psi$ on $W$ we also have that $f-\varepsilon\leq\max(\psi,0)=\varphi$ on $W$. ∎

We will use that the result holds when $V$ is compact [Gub98, Theorem 7.12]. Note that in [Gub98, §7], $K$ was assumed to be algebraically closed, but the argument for [Gub98, Theorem 7.12] does not use this assumption and so we can use the result over any non-archimedean field. Let $f_{+}\coloneqq\max(f,0)$ and $f_{-}\coloneqq\max(-f,0)$ so that $f=f_{+}-f_{-}$. Hence replacing $f$ by $f_{+}$ or $f_{-}$ we can assume that $f\geq 0$.

We can work separately on the connected components of $V$, hence we may assume that $V$ is connected. As in the proof of Proposition 2.8, we can find a locally finite covering $(T_{i})_{i\in I}$ of $V$ made of compact strictly $K$-analytic domains with $I$ finite or countable. In the following, we assume $I={\mathbb{N}}$. The finite case is similar and easier. Applying a compactness argument to the $T_{i}$’s, we can find $(W_{i})_{i\in{\mathbb{N}}}$ and $(U_{i})_{i\in{\mathbb{N}}}$ two locally finite coverings of $V$ by compact strictly $K$-analytic domains of $V$ such that for all $i\in{\mathbb{N}}$ we have $W_{i}\subset U_{i}^{\circ}$.

Let us now fix $\varepsilon>0$ and let us construct a family of piecewise ${\mathbb{Q}}$-linear functions $(\varphi_{i})_{i\in{\mathbb{N}}}$ such that

1. (i)

   for all $i\in{\mathbb{N}}$, ${\rm supp}(\varphi_{i})\subset U_{i}$ and $\varphi_{i}\geq 0$.

2. (ii)

   for all $n\in{\mathbb{N}}$ we have $f\geq\sum_{i=1}^{n}\varphi_{i}\geq f-\varepsilon$ on $\cup_{i=1}^{n}W_{i}$.

3. (iii)

   $f\geq\sum_{i=1}^{n}\varphi_{i}$ on $V$.

Observe that this will conclude the proof of the proposition since then $\varphi\coloneqq\sum_{i\in{\mathbb{N}}}\varphi_{i}$ is a well defined piecewise ${\mathbb{Q}}$-linear function such that $|f-\varphi|\leq\varepsilon$. The rest of the proof is dedicated to construct inductively a family $(\varphi_{i})_{i\in{\mathbb{N}}}$ satisfying the conditions (i), (ii) and (iii).

Let us consider $n\geq 1$ and let us assume that we are given piecewise ${\mathbb{Q}}$-linear functions $\varphi_{1},\ldots,\varphi_{n}$ satisfying the above conditions. We will now construct a piecewise ${\mathbb{Q}}$-linear function $\varphi_{n+1}$ such that $\varphi_{1},\ldots,\varphi_{n+1}$ satisfies the conditions (i), (ii) and (iii).

By the density result in the compact case [Gub98, Theorem 7.12], we know that there exists a piecewise ${\mathbb{Q}}$-linear function $g\colon W_{n+1}\to{\mathbb{R}}$ such that

(2.13.1)

$$f-\sum_{i=1}^{n}\varphi_{i}-\varepsilon\leq g\leq f-\sum_{i=1}^{n}\varphi_{i}\hskip 20.0pt{\rm on}\ W_{n+1}$$

Then by Lemma 2.11 applied to $g$ and $W_{n+1}\subset U_{n+1}\subset V$, there exists a piecewise ${\mathbb{Q}}$-linear function $\Psi\colon V\to{\mathbb{R}}$ which extends $g$ and with ${\rm supp}(\Psi)\subset U_{n+1}$. Then (2.13.1) becomes

(2.13.2)

$$f-\varepsilon\leq\Psi+\sum_{i=1}^{n}\varphi_{i}\leq f\hskip 20.0pt{\rm on}\ W_{n+1}.$$

Then we set

$$\psi\coloneqq\max(0,\Psi).$$

From this definition, we get that ${\rm supp}(\psi)\subset{\rm supp}(\Psi)\subset U_{n+1}$. It is a piecewise ${\mathbb{Q}}$-linear function by Proposition 2.10 (d) and it satisfies $\psi\geq 0$. Now, (2.13.2) combined with the condition (iii) for $n$ yields

(2.13.3)

$$\psi+\sum_{i=1}^{n}\varphi_{i}\leq f\hskip 20.0pt\ {\rm on}\ W_{n+1}.$$

Also, since $\Psi\leq\psi$, we deduce from (2.13.2) that

(2.13.4)

$$f-\varepsilon\leq\psi+\sum_{i=1}^{n}\varphi_{i}\hskip 20.0pt{\rm on}\ W_{n+1}.$$

On the other hand, since $\psi\geq 0$, the condition (ii) for $n$ yields

(2.13.5)

$$f-\varepsilon\leq\psi+\sum_{i=1}^{n}\varphi_{i}\ \hskip 20.0pt\ {\rm on}\ \bigcup_{i=1}^{n}W_{i}.$$

From (2.13.2), (2.13.3), (2.13.4) and (2.13.5), we deduce that

(2.13.6)

$$f-\varepsilon\leq\psi+\sum_{i=1}^{n}\varphi_{i}\leq f\hskip 20.0pt\ {\rm on}\ \bigcup_{i=1}^{n+1}W_{i}.$$

Lemma 2.12 applied to the non negative function $f-\sum_{i=1}^{n}\varphi_{i}\colon V\to{\mathbb{R}}$ and to the compact $K$-analytic domain $\cup_{i=1}^{n+1}U_{i}$ yields a piecewise ${\mathbb{Q}}$-linear function $\chi\colon V\to{\mathbb{R}}$ such that $\chi\geq 0$ and

(2.13.7)

$$f-\sum_{i=1}^{n}\varphi_{i}-\varepsilon\leq\chi\leq f-\sum_{i=1}^{n}\varphi_{i}\hskip 20.0pt\ {\rm on}\ \bigcup_{i=1}^{n+1}U_{i}.$$

We then set

$$\varphi_{n+1}\coloneqq\min(\psi,\chi).$$

By Proposition 2.10 (d), $\varphi_{n+1}$ is a piecewise ${\mathbb{Q}}$-linear function. Since $\psi\geq 0$ and $\chi\geq 0$ we get that $\varphi_{n+1}\geq 0$ and we also get that for $x\in V$, $\psi(x)=0\Rightarrow\varphi_{n+1}(x)=0$. This implies that ${\rm supp}(\varphi_{n+1})\subset{\rm supp}(\psi)\subset U_{n+1}$. Hence (i) is satisfied for $\varphi_{n+1}$.

Let us now prove that

(2.13.8)

$$\sum_{i=1}^{n+1}\varphi_{i}\leq f\ \ {\rm on}\ V.$$

Let $x\in V$. We first suppose that $x\in U_{n+1}$. Then by (2.13.7), we have $\chi(x)+\sum_{i=1}^{n}\varphi_{i}(x)\leq f(x)$. By definition of $\varphi_{n+1}$, we have $\varphi_{n+1}\leq\chi$ hence

$$\varphi_{n+1}(x)+\sum_{i=1}^{n}\varphi_{i}(x)\leq\chi(x)+\sum_{i=1}^{n}\varphi_{i}(x)\leq f(x).$$

If $x\notin U_{n+1}$, then we have $\psi(x)=0$ since ${\rm supp}(\psi)\subset U_{n+1}$, hence $\varphi_{n+1}(x)=0$. So by the condition (iii) for $n$, we get

$$\sum_{i=1}^{n+1}\varphi_{i}(x)=\sum_{i=1}^{n}\varphi_{i}(x)\leq f(x).$$

This proves (2.13.8), whence condition (iii) holds for $n+1$.

Let us finally prove that

$$f-\varepsilon\leq\sum_{i=1}^{n+1}\varphi_{i}\leq f\hskip 30.0pt{\rm on}\ \bigcup_{i=1}^{n+1}W_{i}.$$

The right inequality has been proven in (2.13.8) so it only remains to prove the left inequality. By (2.13.6), we have

(2.13.9)

$$f-\varepsilon\leq\psi+\sum_{i=1}^{n}\varphi_{i}\hskip 30.0pt{\rm on}\ \bigcup_{i=1}^{n+1}W_{i}$$

and by construction (see (2.13.7) having in mind that $W_{i}\subset U_{i}$), we have

(2.13.10)

$$f-\varepsilon\leq\chi+\sum_{i=1}^{n}\varphi_{i}\hskip 30.0pt{\rm on}\ \bigcup_{i=1}^{n+1}W_{i}.$$

Hence (2.13.9) and (2.13.10) yield that

$$f-\varepsilon\leq\min(\psi,\chi)+\sum_{i=1}^{n}\varphi_{i}=\sum_{i=1}^{n+1}\varphi_{i}\hskip 20.0pt{\rm on}\ \bigcup_{i=1}^{n+1}W_{i}$$

which proves condition (ii) for $\varphi_{1},\ldots,\varphi_{n+1}$. By induction, this proves the existence of a family $(\varphi_{i})_{i\in{\mathbb{N}}}$ satisfying conditions (i), (ii) and (iii). ∎

The argument in [Gub98, Lemma 7.6] shows that $L$ admits a formal metric. Hence, tensoring by $L^{-1}$, we can assume that $L=\mathcal{O}_{V}$ and we are reduced to prove that for any continuous function $f\colon V\to{\mathbb{R}}$ there exists a sequence of model functions $(\varphi_{n})_{n\in{\mathbb{N}}}$ which converges uniformly to $f$ which was done in Proposition 2.13. ∎

It follows from [Ber93, Theorem 1.6.1] that the base change of $V$ to $F$ is a paracompact strictly $F$-analytic space. Property (a) is obvious.

To prove (b), we assume that ${\|\hskip 4.30554pt\|}$ is a piecewise linear metric on $L\hat{\otimes}_{K}F$. We have seen in 2.2 that $(V,L)$ has a formal ${K^{\circ}}$-model $({\mathfrak{V}},{\mathfrak{L}})$ and so we may assume that $L={\mathcal{O}}_{V}$ by passing to ${\|\hskip 4.30554pt\|}/{\|\hskip 4.30554pt\|}_{{\mathfrak{L}}\hat{\otimes}_{{K^{\circ}}}F^{\circ}}$. By Proposition 2.8, there is a formal $F^{\circ}$-model $({\mathfrak{V}}^{\prime\prime},{\mathfrak{L}}^{\prime\prime})$ of $(V\hat{\otimes}_{K}F,L\hat{\otimes}_{K}F)$ such that ${\|\hskip 4.30554pt\|}={\|\hskip 4.30554pt\|}_{{\mathfrak{L}}^{\prime\prime}}$. By Raynaud’s theorem [BL93a, Theorem 4.1], we may assume that there is an admissible formal blowing up ${\mathfrak{V}}^{\prime\prime}\to{\mathfrak{V}}\hat{\otimes}_{{K^{\circ}}}F^{\circ}$. Note that $L={\mathcal{O}}_{V}$ yields that ${\mathfrak{L}}^{\prime\prime}={\mathcal{O}}(E)$ for a vertical Cartier divisor $E$ on ${\mathfrak{V}}^{\prime\prime}$. Replacing ${\|\hskip 4.30554pt\|}$ by a suitable multiple, we may assume that $E$ is an effective Cartier divisor.

An approximation argument based on the density of the algebraic closure of $K$ in $F$ shows that the coherent ideal of the admissible formal blowing up and hence the formal model ${\mathfrak{V}}^{\prime\prime}$ are defined on a formal $(K^{\prime})^{\circ}$-model ${\mathfrak{V}}^{\prime}$ for a finite subextension $K^{\prime}/K$ of $F/K$. We choose a finite covering $({\mathfrak{U}}_{i}^{\prime})_{i\in I}$ of ${\mathfrak{V}}^{\prime}$ by formal affine open subsets ${\mathfrak{U}}_{i}^{\prime}$ of ${\mathfrak{V}}^{\prime}$. Then the coherent sheaf of ideals ${\mathcal{O}}(-E^{\prime\prime})$ restricted to ${\mathfrak{U}}_{i}^{\prime}\hat{\otimes}_{(K^{\prime})^{\circ}}F^{\circ}$ is generated by finitely many regular functions. A similar approximation argument as above shows that all these generators can be replaced by regular functions on ${\mathfrak{U}}_{i}^{\prime}$ if we replace $K^{\prime}$ by a larger finite subextension of $F/K$. We conclude that ${\mathfrak{L}}^{\prime\prime}={\mathcal{O}}(E)$ is defined on ${\mathfrak{V}}^{\prime}$ proving (b). Note that uniqueness is obvious. ∎