Lemma 8

Let B be a bounded subset of $\mathscr{D}(\mathbf{R}^n)$. The topology induced on B by the topology of $\mathscr{S}(\mathbf{R}^n)$ coincides with the topology induced by $\mathscr{D}(\mathbf{R}^n)$.

Since the inclusion of $\mathscr{D}(\mathbf{R}^n)$ in $\mathscr{S}(\mathbf{R}^n)$ is continuous, the topology on B induced by $\mathscr{D}(\mathbf{R}^n)$ is finer than that induced by $\mathscr{S}(\mathbf{R}^n)$. On the other hand, there exists a compact subset K of $\mathbf{R}^n$ such that $B\subset \mathscr{C}_K^{\infty}(\mathbf{R}^n)$. For every $\alpha \in \mathbf{N}^n$, one then has $p_{\alpha ,K}(\varphi )\leqslant \widetilde{q}_{\alpha ,0}(\varphi )$, which implies that the topology induced by $\mathscr{S}(\mathbf{R}^n)$ is finer than that induced by $\mathscr{D}(\mathbf{R}^n)$.