Example

The function $\gamma_n$ defined on $\mathbf{R}^n$ by $\gamma_n(x) =$ exp($-\|x\|^2$) belongs to $\mathscr{S}(\mathbf{R}^n)$. In fact, one proves by induction on $k$ that, for every integer $k\in \mathbf{N}$, there exists a polynomial $P_k\in \mathbf{R}[X]$ such that $\partial_k\gamma_1= P_k\gamma_1$.

For all $\alpha = (\alpha_i)\in \mathbf{N}^n$ and $\beta = (\beta_i)\in \mathbf{N}^n$, and every $x= (x_i)\in \mathbf{R}^n$, one then has

which is a bounded quantity when $x$ varies in $\mathbf{R}^n$.

Let $\alpha \in \mathbf{N}^n$ and $\beta \in \mathbf{N}^n$. If $\varphi \in \mathscr{S}(\mathbf{R}^n)$, then $X^{\beta}\partial^{\alpha}(\varphi )$ is again a Schwartz function; the resulting mapping $\varphi \mapsto X^{\beta}\partial^{\alpha}\varphi$ of $\mathscr{S}(\mathbf{R}^n)$ into itself is continuous.

The space $\mathscr{S}(\mathbf{R}^n)$ is a topological algebra. More precisely, if $\varphi_1$ and $\varphi_2$ belong to $\mathscr{S}(\mathbf{R}^n)$, then $\varphi_1\varphi_2$ is a Schwartz function such that

$0\leqslant \gamma \leqslant \alpha$

for all $\alpha$ and $\beta \in \mathbf{N}^n$ (prop. 1 of IV, p. 196).

The canonical inclusion of $\mathscr{S}(\mathbf{R}^n)$ in the space $\mathscr{C}^{\infty}(\mathbf{R}^n)$, endowed with the topology described in No.$^o3$ of IV, p. 200, is continuous, since

sup$_{x\in K}|\partial^{\alpha}\varphi (x)|\leqslant q_{\alpha ,0}(\varphi )$

for every compact subset K of $\mathbf{R}^n$, every $\alpha \in \mathbf{N}^n$ and every Schwartz function $\varphi$.