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?
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.