Proposition 14

Let $f\in \mathscr{C}^{\infty}(\mathbf{R}^n)$. Suppose that for every $\alpha$ in $\mathbf{N}^n$, the function $\partial^{\alpha}f$ has polynomial growth. The linear mapping of the space $\mathscr{S}(\mathbf{R}^n)$ into itself defined by $\varphi \mapsto f \varphi$ is continuous.

For every $\varphi$ in $\mathscr{S}(\mathbf{R}^n)$, the function $f \varphi$ belongs to $\mathscr{C}^{\infty}(\mathbf{R}^n)$. By hypothesis, for every $\alpha \in \mathbf{N}^n$, there exist an integer $k_{\alpha}\geqslant 0$ and a real $C_{\alpha}$ such that $|\partial^{\alpha}f(x)|\leqslant C_{\alpha}(1 +\|x\|)^{k_{\alpha}}$ for every $x$ in $\mathbf{R}^n$. Let $\varphi \in \mathscr{S}(\mathbf{R}^n)$. Let $\alpha \in \mathbf{N}^n$ and $k\in \mathbf{N}$. By Prop. 1 of IV, p. 196, there follows

$\widetilde{q}_{\alpha ,k}(f \varphi )\leqslant \sum$ sup $\|x\||\partial f(x)\partial \varphi (x)|$

$0\leqslant \beta \leqslant \alpha$

$\leqslant \sum C_{\beta}$ sup $\|x\|(1 +\|x\|)^{^{\beta}}|\partial \varphi (x)|$.

$0\leqslant \beta \leqslant \alpha$

Let $\beta \in \mathbf{N}^n$ be such that $0\leqslant \beta \leqslant \alpha$. For every $x\in \mathbf{R}^n$, one has

$\|x\|^k(1 +\|x\|)^{k_{\beta}}|\partial^{\alpha-\beta}\varphi (x)|\leqslant$ sup$_{\|x\|\leqslant 1}2^{k_{\beta}}|\partial^{\alpha-\beta}\varphi (x)|$

+ sup$_{x\in\mathbf{R}^n}2^{k_{\beta}}\|x\|^{k+k_{\beta}}|\partial^{\alpha-\beta}\varphi (x)|$ whence finally

$0\leqslant \beta \leqslant \alpha$

which implies the proposition.