Let $\mathcal S'(\mathbb R^n)$ the space of all continuous linear functions from the Schwartz space $\mathcal S(\mathbb R^n)$ to $\mathbb C$, and $\mathcal D'(\mathbb R^n)$ the space of all continuous linear functions from the space $\mathcal D(\mathbb R^n)$ of the smooth functions with compact support $\mathcal D(\mathbb R^n)$ to $\mathcal C$.
If ${f_j},f \in {\mathcal S}'({\mathbb R}^n)$ and ${f_j} \to f$ in ${\mathcal D}'(\mathbb R^n)$, is it necessarily that ${f_j} \to f$ in ${\mathcal S}'(\mathbb R^n)$?