Let $\phi\in\mathscr{D}$. Then $f\phi\in\mathscr{D}$ for every smooth function $f$.

Question

Let $\phi\in\mathscr{D}$, where $\phi$ is a test function and $\mathscr{D}$ is the set of all test functions. Then $f\phi\in\mathscr{D}$ for every smooth function $f$.

This one seems...trivial. So many of the other problems I've worked through rely on this. I'm not sure how to go about this, or where to even start.

Answer 1

Observe that $\mathrm{Supp}(f\phi)\subset\mathrm{Supp}(\phi)$. Since $\mathrm{Supp}(\phi)$ is compact and $\mathrm{Supp}(f\phi)$ is closed, $\mathrm{Supp}(f\phi)$ is compact. Furthermore, the smoothness of $f\phi$ comes from the product rule of differentiation.