I'm reading Gelbart's Introduction to the Selberg Trace Formula https://arxiv.org/abs/math/0407288. In his paper he seems to have used the consequence that a smooth function with compact support is a Schwartz-Bruhat function. I'm wondering whether this is correct.
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$\begingroup$ Certainly this is correct on adelic vector spaces. Of course, at archimedean places the (local) Schwartz space is larger. Also, the "locally constant" has to refer to the adele topology, since we would exclude things involving any infinite product of characteristic functions of $p\mathbb Z_p$, for example. $\endgroup$– paul garrettCommented Jul 2, 2017 at 21:46
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