Restricted tensor product of local test-function spaces

Let $F$ be a number field, $F_v$ its completions, and $\mathcal S(F_v)$ the local Schwartz–Bruhat spaces. Put $F_\infty=\prod_{v\mid\infty}F_v$ , and let $\mathcal S(F_\infty)$ be the completed nuclear tensor product of the archimedean local Schwartz spaces. For each finite place set $e_v=1_{\mathcal O_v}$ . The restricted tensor product of the local test-function spaces is

\mathcal S(F_\infty)\,\widehat\otimes\! \bigotimes_{v\nmid\infty}'\mathcal S(F_v) =\underset{S}{\mathop{\mathrm{colim}}}\, \left(\mathcal S(F_\infty)\widehat\otimes \bigotimes_{v\in S}\mathcal S(F_v)\right) \otimes\left(\bigotimes_{\substack{v\nmid\infty\\v\notin S}} \mathbb C e_v\right),

where $S$ ranges over finite sets of finite places. The vectors $e_v$ are part of the definition.

Canonical realization

An elementary tensor determines a function on the restricted product $\mathbb A_F=\prod_v'F_v$ by

x=(x_v)_v\longmapsto\prod_v f_v(x_v).

Only finitely many factors differ from $1_{\mathcal O_v}$ , and an adele lies in $\mathcal O_v$ at almost every finite place, so this product is well defined. Finite sums of completely separated local functions realize a dense algebraic subspace of the adelic test-function space. They need not exhaust $\mathcal S(F_\infty)$ when there is more than one archimedean place.

Topology

For a fixed finite $S$ , the unrestricted factors form a completed nuclear test-function tensor product, while factors outside $S$ remain fixed. The adelic Schwartz topology is the standard locally convex inductive-limit (LF) topology over these finite sets. Completely separated elementary tensors are dense in each archimedean completion and hence in the resulting adelic space.

Dependence on distinguished vectors

A restricted tensor product is not determined by the spaces $\mathcal S(F_v)$ alone; the vectors $e_v$ are part of its data. For the additive adeles, $1_{\mathcal O_v}$ is the standard unramified choice. Replacing finitely many $e_v$ gives the same restricted tensor-product space up to its evident canonical identification, whereas changing infinitely many can produce a different restricted product.

References

André Weil, Basic Number Theory, 2nd ed., Springer, 1973. DOI record. Relevant: Chapter II, “Adeles,” restricted products and adelic test functions.
Daniel Bump, Automorphic Forms and Representations, Cambridge University Press, 1997. DOI record. Relevant: Chapter 3, restricted tensor products in adelic harmonic analysis.