Surjectivity of a restriction map on distributions I'm reading Kudla's exposition of Tate's thesis in the book "An Introduction to the Langlands Program" and have gotten stuck on some analytic details. Here's the setup: let $F$ be $\mathbb{R}$ or $\mathbb{C}$, so that there is an inclusion $C^{\infty}_c(F^{\times}) \subset \mathcal{S}(F)$, where $\mathcal{S}(F)$ is the Schwartz space of rapidly decreasing functions. Kudla claims that this induces a surjection $\mathcal{S}(F)^{\vee} \to C^{\infty}_c(F^{\times})^{\vee}$ on the respective spaces of distributions, which is not clear to me.
Can we perhaps invoke a form of the Hahn-Banach theorem? But $C^{\infty}_c(F^{\times})$ doesn't have the subspace topology coming from $\mathcal{S}(F)$, does it?
 A: Your misgivings are justified: the test functions do not have the topology from Schwartz functions, and not every distribution (dual of test fcns) is tempered (dual of Schwartz). There is a natural continuous injection because test functions are dense in Schwartz.
Edit: to clarify, and compare to non-archimedean case: tempered distributions do not surject to all distributions, because many/typical distributions are not tempered ("at infinity"). In the non-archimedean case, these two things are identical. When the test functions are required to be supported away from $0$, in the archimedean case the map of tempered to the dual has Dirac delta and derivatives in its kernel, but is still not surjective, because the non-temperedness is a condition at infinity. In the non-archimedean case, the support-away-from-$0$ only puts Dirac delta in the kernel of the map from tempered to the dual, because that's the only distribution supported at a point, in the non-archimedean case.
So the map from tempered distributions to the dual of test functions supported away from $0$ is not surjective, no. E.g., $\sum_{1\le n\in \mathbb Z} e^n\cdot \delta_n$ is a distribution that is not tempered. And the kernel of linear combinations of Dirac delta and derivatives at $0$ does not put this into "tempered", either.
One more time, in symbols: let $V$ be the (ind-finite) space of all linear combinations of Dirac delta and derivatives. Let $S'$ be tempered distributions. Let $X$ be test functions supported away from $0$, and $X'$ its dual. Then, sure, $X\subset S$ with dense image, but/and $S'\rightarrow X'$ factors through $S'/V$, but/and is not surjective. (Not even "modulo $V$").
But surely this mis-statement doesn't really matter to Kudla's argument.
