- Suppose $\{K_n\} \subset \mathcal{D}'$, and $\langle K_n, u\rangle$ converges for every $u$ in a dense subset of $C_0^\infty$, does $K_n$ converges to some $K\in \mathcal{D}'$?
If this is not true, how about:
- Suppose $\{K_n\} \subset \mathcal{D}', K \in \mathcal{D}'$, and $\langle K_n, u\rangle \to \langle K, u\rangle$, for every $u$ in a dense subset of $C^\infty_0$, does $K_n$ converges to $K$?
A proof or a counterexample would be appreciated.