This is an exercise in Rudin's Functional analysis, which is 6.23.
The problem:
$f_i \in L_{loc}^1(\mathbb{R}^n), \ \lim\limits_{i \to \infty}(f_i*\phi)(x)$exists, $\forall \phi \in \mathscr{D}, \ x \in \mathbb{R}^n$
Prove that $D^{\alpha}(f_i*\phi)$ converge uniformly on compact sets, for each multi-index $\alpha$
It is equivalent to prove that $f_i*\phi$ converge in $C^\infty$, then I want to show that it's a Cauchy sequence, but I don't know how to continue. I also think the Banach-Steinhaus can be used, but it can only get the equicontinuity of $f_i$ when we fix $x$.
Thank you very much!