This page is about the space of sections:
Let $E \stackrel{\mathrm{fb}}{\rightarrow} \Sigma$ be a smooth vector bundle. On its real vector space $\Gamma_{\Sigma}(E)$ of smooth sections consider the seminorms indexed by a compact subset $K \subset \Sigma$ and a natural number $N \in \mathbb{N}$ and given by $$ \begin{aligned} \Gamma_{\Sigma}(E) & \stackrel{p_{K}^{N}}{\longrightarrow} [0,\infty)\\ \Phi & \longmapsto \max _{n \leq N}\left(\sup _{x \in K}\left|\left(\nabla^{E}\right)^{n} \Phi(x)\right|\right) \end{aligned} $$ where on the right we have the absolute values of the covariant derivatives of $\Phi$ for any fixed choice of connection on $E$ and norm on the tensor product of vector bundles $\left(T^{*} \Sigma\right)^{\otimes_{\Sigma}^{n}} \otimes_{\Sigma} E$.
Let $(U, \varphi, \rho)$ be a total trivialization triple for $E$.and $u \in \Gamma_{U}(E)$. On $U$ we may write $u=u^{a} s_{a}$
Now in general we have $$\left(\nabla^{E}\right)^{k} u=\left(\left(\nabla^{E}\right)^{k} u\right)_{i_{1} \ldots i_{1}}^{a} d x^{i_{1}} \otimes \cdots \otimes d x^{i_{k}} \otimes s_{a}$$
where $$\left(\left(\nabla^{E}\right)^{k} u\right)_{i_{1} \cdots i_{k}}^{a} \circ \varphi^{-1}=\frac{\partial ^k}{\partial x^{i_{1}} \ldots \partial x^{i_{k}}}\left(u^{a} \circ \varphi^{-1}\right)+\sum_{\underline{n} \mid<k_{l}} \sum_{l=1} C_{\eta l} \partial^{\eta}\left(u^{l} \circ \varphi^{-1}\right)$$
where for each $\eta$ and $l, C_{\eta l}$ is a polynomial in terms of Christoffel symbols (of the linear connection on $M$ and connection in $E$ ).
Now suppose we have a metric $g$ in $U$ and a metric $h$ in $E$ then we have that $$\left|\left(\nabla^{E}\right)^{n} u\right|= \left(\left(\nabla^{E}\right)^{k} u\right)_{i_{1} \cdots i_{k}}^{a}\left(\left(\nabla^{E}\right)^{k} u\right)_{j_{1} \cdots j_{k}}^{b} g^{i_1 j_i}\cdots g^{i_k j_k}h^{ab}$$ How to prove that this space is complete in respect to the seminorm?
Now In this article on page 6 GREEN-HYPERBOLIC OPERATORS ON GLOBALLY HYPERBOLIC SPACETIMES the author says that an Arzelà-Ascoli argument would show this but I don't see how.
If anyone would like me to expand the question please let me know.