Proposition 10

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

Since the inclusion of $\mathscr{S}(\mathbf{R}^n)$ in $\mathscr{C}^{\infty}(\mathbf{R}^n)$ is continuous, it is enough to prove that, for every open subset V of $\mathscr{S}(\mathbf{R}^n)$, the intersection $V\cap B$ is open in B for the topology induced by $\mathscr{C}^{\infty}(\mathbf{R}^n)$.

Let V be an open subset of $\mathscr{S}(\mathbf{R}^n)$. Let $\varphi_0\in V\cap B$. There exist a finite set I, a family $(\alpha_i, k_i)_{i\in I}\in (\mathbf{N}^n\times \mathbf{N})^I$ and a real number $\varepsilon >0$ such that V contains the set of $\varphi \in \mathscr{S}(\mathbf{R}^n)$ satisfying

sup$_{i\in I}\widetilde{q}_{\alpha_i,k_i}(\varphi -\varphi_0)\leqslant \varepsilon$.

Since B is bounded in $\mathscr{S}(\mathbf{R}^n)$, there exists $M>0$ such that the seminorms $\widetilde{q}_{\alpha_i,k_i+1}$ for $i\in I$ are bounded by M on B. Let $\delta >0$ and $T>0$ be real numbers. By inequality (3), as soon as $\varphi \in B$ satisfies the bound

(4) sup$_{i\in I}$ sup$_{\|x\|\leqslant T}|\partial^{\alpha_i}(\varphi -\varphi_0)|\leqslant \delta$,

one has

$_k$ 2M

sup $\widetilde{q}_{\alpha_i,k_i}(\varphi -\varphi_0)\leqslant \delta T$ +.

$_{i\in I}$ T

Let $T =\frac{4M}{\varepsilon}$, then $\delta =\frac{\varepsilon}{2T^k}$. One sees that $V\cap B$ contains the neighbourhood of $\varphi_0$ in B for the induced topology of $\mathscr{C}^{\infty}(\mathbf{R}^n)$ which is defined by (4). This concludes the proof.