Example 3

The set $\mathbf{N}_{\mathbf{R}}=\mathbf{N}\cup \{\infty , \omega \}$ is endowed with the total ordering described in VAR, R2, p. 10, such that $n <\infty < \omega$ for every $n\in \mathbf{N}$. Let $r\in \mathbf{N}_{\mathbf{R}}$. Let $n\in \mathbf{N}$ and let U be an open subset of $\mathbf{R}^n$. Let $k\in \mathbf{N}$ be such that $k\leqslant r$. Let $(n_{\alpha})_{|\alpha|\leqslant k}$ be a family of elements of $\mathscr{C}^r$(U), where the multi-indices considered belong to $\mathbf{N}^n$. The family $(n_{\alpha})$ defines a scalar differential operator D of order $\leqslant k$ on U (cf. VAR, R2, 14.1.6, 14.1.4). For every integer $m$ such that $k\leqslant m\leqslant r$, the differential operator D defines a linear mapping of $C^m(U)$ into $C^{m-k}(U)$ which sends $f\in C^m(U)$ to

The same formula defines a continuous linear mapping of $\mathscr{D}(U)$ into $\mathscr{D}'(U)$ (def. 2 of IV, p. 208).