Example 4

\* Let U be an open set in $\mathbf{R}^n$ and let $\mathscr{C}^\infty(U)$ be the Fréchet space of infinitely differentiable functions on U (III, p. 9). We shall prove that this is a Montel space. Since $\mathscr{C}^\infty(U)$ is a Fréchet space, it is barrelled (III, p. 25, corollary). Let B be a bounded subset of $\mathscr{C}^\infty(U)$ and let K be a compact subset of U. For every $\alpha \in \mathbf{N}^n$ let $H_{\alpha, K}$ be the set of restrictions to K of the functions $\partial^\alpha f$, as $f$ runs through B. Let $\alpha \in \mathbf{N}^n$; for every $\beta \in \mathbf{N}^n$ such that $|\beta| = |\alpha| + 1$, the set $H_{\alpha, K}$ is bounded in $\mathscr{C}(K)$ since B is bounded in $\mathscr{C}^\infty(U)$; by VAR, R., No. 2.2.3, the set $H_{\alpha, K}$ is equicontinuous, hence (GT, X, § 2, No. 5) relatively compact in $\mathscr{C}(K)$. But the topology of $\mathscr{C}^\infty(U)$ is the coarsest among the topologies for which all the maps $f \mapsto \partial^\alpha f|K$ from $\mathscr{C}^\infty(U)$ into $\mathscr{C}(K)$ are continuous, therefore B is relatively compact in $\mathscr{C}^\infty(U)$ (GT, I, § 4, No. 1, prop. 3 and § 9, No. 5, corollary).

Similarly, the space $\mathscr{C}_0^\infty(U)$ of all infinitely differentiable functions with compact support in U (III, p. 9) is a Montel space. For, $\mathscr{C}_0^\infty(U)$ is the strict inductive limit of a sequence $\mathscr{C}_{H_n}^\infty(U)$ of Fréchet spaces (III, p. 9), and it is enough to see that each of the spaces $\mathscr{C}_{H_n}^\infty(U)$ is a Montel space (Example 3). But a bounded and closed subset of $\mathscr{C}_0^\infty(U)$ is closed and bounded in $\mathscr{C}^\infty(U)$, hence compact in $\mathscr{C}^\infty(U)$, and consequently in $\mathscr{C}_{H_n}^\infty(U)$. \*