Let $(V_{i},\varphi_{U_{i}})_{i\in I}$ be finitely many tropical charts contained in $V_{i}$ and let $\Delta_{i}$ be a polytope contained in the open subset $\Omega_{i}:={\rm trop}_{U_{i}}(V_{i})$ of ${\rm Trop}(U_{i})$. We consider the space $A^{p,q}(V_{i},U_{i},\Delta_{i}:i\in I)$ of $(p,q)$-forms $\alpha$ on $W$ with support in $C:=\bigcup_{i\in I}{\rm trop}_{U_{i}}^{-1}(\Delta_{i})$ such that $\alpha$ is given on $V_{i}$ by a superform $\alpha_{i}\in A^{p,q}(\Omega_{i})$ for every $i\in I$. Since the tropicalization map is proper (see 4.4), the set $C$ is compact. Similarly as in the complex case, we endow $A^{p,q}(V_{i},U_{i},\Delta_{i}:i\in I)$ with the structure of a locally convex space such that a sequence $\alpha_{k}$ converges to $\alpha$ if and only if all derivatives of the superforms $\alpha_{k,i}$ converge uniformly to the derivatives of the superform $\alpha_{i}$ on $\Delta_{i}$. Here, $\alpha_{k,i}$ (resp. $\alpha_{i}$) is the superform on $\Omega_{i}$ which defines $\alpha_{k}$ (resp. $\alpha$) on $V_{i}$ and we mean more precisely the derivatives of the coefficients of $\alpha_{k,i}|_{\Delta_{i}}$ (resp. $\alpha_{i}|_{\Delta_{i}}$). It follows easily from Proposition 4.16 that $A_{c}^{p,q}(W)$ is the union of all spaces $A^{p,q}(V_{i},U_{i},\Delta_{i}:i\in I)$ with $(V_{i},U_{i},\Delta_{i}:i\in I)$ ranging over all possibilities as above.