Example 4

Let X be a differential manifold of class $C^{\infty}$ over $\mathbf{R}$ and let $\mathscr{C}^{\infty}(X;\mathbf{R})$ be the sheaf on X of numerical functions of class $C^{\infty}$. If P is a differential operator with $C^{\infty}$ coefficients on X, the family of the restrictions of P to the open sets of X is a morphism of the sheaf $\mathscr{C}^{\infty}(X;\mathbf{R})$ into itself. One can prove that conversely, every $\mathbf{R}$-linear morphism of the sheaf $\mathscr{C}^{\infty}(X;\mathbf{R})$ into itself is locally of this form (I, p. 142, exerc. 3).