Proposition 10

Suppose that the vector bundle E is locally of finite rank. Let $s$ be an integer such that $0\leqslant s < r$ and Y a relatively compact open subset of X. The linear mapping $f\mapsto f|Y$ of $\mathscr{S}^r(X; E)$ into $\mathscr{S}^s(Y; E)$ is compact.

Suppose first that there exist an integer $n\geqslant 0$ and a finite-dimensional 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 spaces $\mathscr{S}^r(X; E)$ and $\mathscr{S}^s(Y; E)$ are isomorphic to $\mathscr{C}^r(X; F)$ and $\mathscr{C}^s(Y; F)$, respectively, and prop. 10 is a consequence of prop. 8.

Let us pass to the general case. Let C be a compact subset of X containing Y. For every point $x$ of C, choose an open neighbourhood $U(x)$ of $x$ in X which is a coordinate domain over which the vector bundle E is trivializable. Moreover, choose an open relatively compact neighbourhood $V(x)$ of $x$ in $U(x)$. Since the set C is compact, it is covered by a finite family $(V(x_1), . . . ,V(x_m))$ of such open sets. For every $i$, put $U_i= U(x_i)$ and $Y_i= V(x_i)\cap Y$. Then $Y_i$ is an open relatively compact subset of $U_i$ and one has $Y = Y_1\cup \cdots \cup Y_m$. The linear mappings $f\mapsto f|U_i$ of $\mathscr{S}^r(X; E)$ into $\mathscr{S}^r(U_i; E)$ are continuous and the linear mappings $g\mapsto g|Y_i$ of $\mathscr{S}^r(U_i; E)$ into $\mathscr{S}^s(Y_i; E)$ are compact by the first part of the proof. The linear mappings $f\mapsto f|Y_i$ of $\mathscr{S}^r(X; E)$ into $\mathscr{S}^s(Y_i; E)$ are therefore compact (prop. 3 of III, p. 5). Taking into account lemma 4, applied to the covering of Y by the open sets $Y_i$, and remarks 5 and 6 of III, p. 3, the linear mapping $f\mapsto f|Y$ of $\mathscr{S}^r(X; E)$ into $\mathscr{S}^s(Y; E)$ is compact, which completes the proof.