Let $\mathbb{K}$ denote either $\mathbb{R}$ or $\mathbb{C}$. Let $E$ be a $\mathbb{K}$ vector bundle of rank $r$ over a compact oriented Riemannian manifold $(M, g)$ of dimension $n$. There are the following two ways to equip the space $\Gamma(E)$ of sections of $E$ with an $L^2$-norm. First, you can equip the bundle $E$ with a fibre metric $\langle \cdot, \cdot \rangle$, and then define $$\lVert u \rVert = \int_M \lvert u \rvert \operatorname{vol}_g$$ where $\lvert u \rvert = \sqrt{\langle u, u \rangle}$. The second way is choose a cover of $M$ by coordinate charts $x_\ell \colon U_\ell \to \hat{U}_\ell \subset \mathbb{R}^n$ over which $E$ trivialises $E\rvert_{U_\ell} \cong U_\ell \times \mathbb{K}^r$ for $\ell = 1, \ldots, N$, and choose a partition of unity $\chi_1, \ldots, \chi_N$ subordinate to $U_1, \ldots, U_N$. Then given $u \in \Gamma(E)$, we can write $u = \sum_\ell u_\ell$ where $u_\ell = \chi_\ell u$ can be viewed as a smooth function $\hat{U}_\ell \to \mathbb{K}^r$, and we set $$\lVert u \rVert' = \sum_\ell \lVert u_\ell \rVert$$ where $\lVert u_\ell \rVert$ is the usual $L^2$-norm on functions $\hat{U}_\ell \to \mathbb{K}^r$.
It is commonly said in the literature that the norms $\lVert \cdot \rVert$ and $\lVert \cdot \rVert'$ defined in this way are equivalent (for example in Chapter 3 of Lawson and Michelsohn's Spin Geometry book), but I don't see how to show this. I tried writing out the integral $\int_M \lvert u \rvert \operatorname{vol}_g$ out using the partition of unity, but that did not lead me anywhere useful.