Is Schwartz space closed under multiplication and convolution?

Question

I think the both answer is "yes".

How can I prove that Schwartz space is closed under multiplication ?

Because if I know that, it is easy to see that being closed under convolution is satisfied.

Help me please.

Thanks

Answer 1

Hint: If $f, g \in \mathscr S$ and $\alpha$ and $\beta$ are multi-indices, what is $x^\alpha D^\beta(f g)$?

Answer 2

I have found a answer to a similar question, and this answer is due to Davide, not me. I remark that, with some obvious modification, his proof can be applied to this question.

This is the link: Schwartz Space is closed under differentiation and multiplication by polynomials.