Corollary

Let $(\varphi_m)_{m\in\mathbf{N}}$ be a bounded sequence in $\mathscr{S}(\mathbf{R}^n)$. For every function $\varphi \in \mathscr{S}(\mathbf{R}^n)$, the following assertions are equivalent:

a) The sequence $(\varphi_m)$ converges to $\varphi$ in $\mathscr{S}(\mathbf{R}^n)$;

b) The sequence $(\varphi_m)$ converges to $\varphi$ in $\mathscr{C}^{\infty}(\mathbf{R}^n)$.