A Bruhat-Schwarz function on the adele ring $\mathbb{A}_{\mathbb{Q}}$ is defined to be a finite linear combination of characteristic functions of the sets of the form $U_\infty \times \prod_p U_p$ where $U_v \subset \mathbb{Q}_v$ is an open subset and $U_p = \mathbb{Z}_p$ for all but finitely many $p$. The collection of such functions is denoted by $\mathcal{S}(\mathbb{A}_{\mathbb Q})$.
I'd like to know the definition of $\mathcal{S}(\mathbb{A}_{\mathbb Q}^n)$, and more importantly $\mathcal{S}(\mathbb{A}_{\mathbb{Q},f}^n \backslash \{0\} )$ where $\mathbb{A}_{\mathbb Q,f}$ is the ring of finite adeles of $\mathbb{Q}$. (I encountered $\mathcal S(\mathbb{A}_{\mathbb Q,f}^n \backslash \{0\})$ while reading the following paper.)
Or perhaps is there a general definition of a Bruhat-Schwarz function on any topological space?