Schwartz-Bruhat space ($\mathcal{S}(F)$)

For a non-Archimedean local field $F$, the Schwartz-Bruhat space $\mathcal{S}(F)$ consists of the locally constant, compactly supported complex-valued functions on $F$.

A Schwartz-Bruhat function on a non-Archimedean local field is a locally constant, compactly supported complex-valued function, or an element of the Schwartz-Bruhat space. A sufficiently small open additive subgroup leaves it invariant under translation, and only finitely many cosets meet its support.