Proposition 9

The space $\mathscr{S}^r(X; E)$ is complete.

Suppose first that there exist an integer $n\geqslant 0$ and a Banach space F such that X is an open subset of $\mathbf{R}^n$ and E the trivial vector bundle $X\times F$ with base X and fibre F. In this case, the topological vector space $\mathscr{S}^r(X; E)$ is isomorphic to $\mathscr{C}^r(X; F)$, and the result follows from lemma 3 of III, p. 34.

In the general case, let $\mathscr{U}$ be an open covering of X by chart domains such that for every $U\in \mathscr{U}$, the restriction of E to U is trivializable. By the above lemma, the space $\mathscr{S}^r(X; E)$ is isomorphic to the image of the linear mapping $f\mapsto (f|U)_{U\in\mathscr{U}}$, which is closed in the product of the spaces $\mathscr{S}^r(U; E)$ for U in $\mathscr{U}$. By the preceding case, each of the spaces $\mathscr{S}^r(U; E)$ is complete, hence their product is complete (TG, II, p. 17, prop. 10). Consequently $\mathscr{S}^r(X; E)$ is complete.