Remark

Let $\mathscr{U}$ be an open covering of X by chart domains $c_U= (U, \varphi_U,\mathbf{R}^{n_U})$ such that for every $U\in \mathscr{U}$, the restriction of E to U is trivializable of type $F_U$, where $F_U$ is a Banach space. For every $U\in \mathscr{U}$, a section $f$ of E over U is identified with a mapping $f_U$ of $\varphi_U(U)$ into $F_U$. It follows from the proof of prop. 9 that the topology of the space $\mathscr{S}^r(X; E)$ is defined by the family of semi-norms $p_{U,C,\alpha}$ such that

$p_{U,C,\alpha}(f) =$ sup$_{x\in C}\|(\partial^{\alpha}f_U)(x)\|$, where U ranges over $\mathscr{U}, C$ ranges over the set of compact subsets of $\varphi_U(U)$ and $\alpha \in \mathbf{N}^{n_U}$ ranges over the set of multi-indices such that $|\alpha |\leqslant r$.