Timeline for Let $u_n\in\mathcal{D}'(\mathbb{R}^n)$ have $u_n\to0$ where $u_n\in C_c^\infty$ have uniformly compact support. Does $u_n\to0$ in $C_c^\infty$?

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Oct 12, 2017 at 16:57	comment	added	Nate Eldredge		So use the closed graph theorem on $\phi \mapsto u \ast \phi$ as a linear operator from $C^\infty_c(B) \to C^\infty_c(B')$ with the aforementioned Fréchet topologies, to conclude $u \ast \phi_n \to 0$, uniformly with all derivatives and all supported in $B'$, hence $u \ast \phi_n \to 0$ in $\mathcal{D}(\mathbb{R})$. Finally, $B$ was arbitrary. No fancier versions needed this way, unless I missed something.
Oct 12, 2017 at 16:55	comment	added	Nate Eldredge		I was thinking of the fact that if you take a ball $B$, and endow $C^\infty_c(B)$ with the topology generated by uniform seminorms on the derivatives (i.e. a sequence converges iff all derivatives converge uniformly, but not necessarily supported in a single compact set), then this space is Fréchet. $\mathcal{D}(\mathbb{R})$ is the union of such spaces. Now if $\phi_n \to 0$ in $\mathcal{D}(\mathbb{R})$, then all are supported in some ball $B$, and hence all the $u \ast \phi_n$ are supported in some larger ball $B'$.
Oct 12, 2017 at 14:18	history	edited	Abdelmalek Abdesselam	CC BY-SA 3.0	fixed typo
Oct 12, 2017 at 5:29	review	First posts
Oct 12, 2017 at 5:30
Oct 12, 2017 at 5:28	history	answered	saone	CC BY-SA 3.0