Suppose that $X$ is a compact (smooth) $n$-manifold and $E \to X$ be a rank $N$ smooth (complex) vector bundle. Choose finite covering $(U_i)_i$ by domains of the charts $\varphi:U_i \to V_i \subset \mathbb{R}^n$ and let $h_i:E|_{U_i} \to V_i \times \mathbb{C}^N$ be a collection of trivialisations (composed with $\varphi_i \times id$ in order to land in the product $V_i \times \mathbb{C}^N$). Let $h_{i*}$ be a induced map $h_{i*}\xi=h_i \circ \xi \circ \varphi_i^{-1}$ acting from $\Gamma^{\infty}(U_i,E|{U_i})$ to $\Gamma^{\infty}(V_i,\mathbb{C}^N)$ which can be indentified with $[C^{\infty}(V_i)]^N$. Let $(f_i)_i$ be a smooth partition of unity subordinate to $(U_i)_i$ and put $$||\xi||_{s,E}:=\Sigma_i||h_{i*}(f_i\xi)||_{s,\mathbb{R}^n}$$ where on the right hand side we have usual Sobolev $s$-norm given by $\Big(\int_{\mathbb{R}^n}|\hat{f}(y)|^2(1+|y|^2)^sdy\Big)^{\frac{1}{2}}$. In this definition we made three non canonical choices: of coordinates, of trivialisations and of partition of unity. I would therefore like to ask
Why these norms are equivalent for different choices of coordinates/trivialisations/partition of unity?
Moreover, for $s=0$ we can consider the scalar product $$(\xi,\eta):=\int_X\langle\xi(x),\eta(x) \rangle d\mu(x)$$ as an integral with respect to the Riemannian density $d\mu$ and $\langle \cdot,- \rangle$ is a Hermitian metric on $E$.
Why the norm coming from this calar product is equivalent to the norm introduced above with $s=0$?
This question is motivated by the considerations in Chapter IV of Wells' book "Differential Analysis on Complex Manifolds" where details are left to the reader. However I don't see why these questions have to be easy.
