Hello,

Does anyone know a reference or proof of the "if" part of the following statement?

$$ \mu\in \mathcal{S}'(R)\quad\text{if and only if}\quad \mu*\alpha\in\mathcal{S}(R),\forall \alpha\in C_c^\infty(R), $$

where $\mathcal{S}'(R)$ is the space of Schwartz distributions, and $\mathcal{S}(R)$ is the space of smooth function with rapid decrease. I noticed that in Schwartz's book (1966) there is a proof. I am not good at french. I am wondering whether anyone knows any other references about this matter? It may use the Banach–Steinhaus theorem in certain way.

By the way, whether can we improve the above results by asking $\alpha\in\mathcal{S}(R)$?

Thank you very much for any hints. :-)

Anand