Lemma 4

Let $\mathscr{U}$ be an open covering of X. The mapping $u:f\mapsto (f|U)_{U\in\mathscr{U}}$ of $\mathscr{S}^r(X; E)$ into $\prod_{U\in\mathscr{U}}\mathscr{S}^r(U; E)$ is linear, injective, continuous and strict. Its image is closed.

The mapping $u$ is linear, injective and continuous. By virtue of TG, IX, p. 43, prop. 1 and p. 48, cor. 1, every compact subset of X has a finite covering $(C_i)_{i\in I}$ where, for each $i\in I$, the set $C_i$ is a compact subset of one of the open sets of the covering $\mathscr{U}$. It follows that the linear mapping $u$ is strict. Finally, its image is closed, since it consists of the families $(f_U)_{U\in\mathscr{U}}$ such that $f_U$ and $f_V$ coincide in $U\cap V$ for all U and V in $\mathscr{U}$.