For concreteness, let me just take the adeles of $\mathbb{Q}$. Let us call a function $f$ on $\mathbb{A}_{\mathbb{Q}}$ elementary or factorizable if it can be written as $$ f(x_{\infty},x_2,x_3,x_5,\ldots)=g(x_{\infty})\times \prod_{p\ {\rm prime}} g_{p}(x_p) $$ with
- $g_{\infty}$ in the usual Schwartz space of $\mathbb{R}$, i.e., smooth with fast decay at infinity and likewise for all its derivatives.
- for all $p$, $g_p$ is in the Schwartz-Bruhat space for $\mathbb{Q}_p$, i.e., locally constant and compactly supported
- for all except at most finitely many $p$'s, $g_p$ is the indicator function of $\mathbb{Z}_p$.
The space $S(\mathbb{A}_{\mathbb{Q}})$ of Schwartz-Bruhat functions on the adele ring $\mathbb{A}_{\mathbb{Q}}$ is the space of all finite linear combinations of elementary functions as above.