Lemma 3

Let A be the set of multi-indices $\alpha \in \mathbf{N}^n$ such that $|\alpha |\leqslant r$ and let $u$ be the linear mapping $f\mapsto (\partial^{\alpha}f)_{\alpha\in A}$ of $\mathscr{C}^r(U; F)$ into $\mathscr{C}(U; F)^A$.

a) The mapping $u$ is injective, continuous, strict, and its image is closed.

b) The topological vector space $\mathscr{C}^r(U; F)$ is complete.

The mapping $u$ is linear and injective. It is continuous and strict by definition of the topology of $\mathscr{C}^r(U; F)$.

Let $(g_{\alpha})_{\alpha\in A}$ be a point of $\mathscr{C}(U; F)^A$ adherent to the image of $u$. There exists a filter $\mathfrak{F}$ on $\mathscr{C}^r(U; F)$ such that one has $g_{\alpha}=$ lim$_{f,\mathscr{F}}\partial^{\alpha}f$ in $\mathscr{C}(U; F)$ for every $\alpha \in A$. Let $m$ be an integer such that $0\leqslant m\leqslant r$. Reasoning by induction on $m$, one deduces from Theorem 1 of FVR, II, p. 2, that the mapping $g_0$ is of class $C^m$ and that one has $g_{\alpha}=\partial^{\alpha}g_0$ for every $\alpha \in \mathbf{N}^n$ with $|\alpha |\leqslant m$. Thus $g_0$ belongs to the space $\mathscr{C}^r(U; F)$ and its image under $u$ is $(g_{\alpha})_{\alpha\in A}$. This proves that the image of $u$ is closed, and therefore establishes assertion a).

Assertion a) implies that the space $\mathscr{C}^r(U; F)$ is isomorphic to its image in $\mathscr{C}(U; F)^A$; since this image is closed and the space $\mathscr{C}(U; F)^A$ is complete (TG, X, p. 9, Corollary 3 of Theorem 2 and TG, II, p. 17, Proposition 10), the space $\mathscr{C}^r(U; F)$ is complete.