Let $(\mathcal{M},g)$ be a (pseudo)-Riemannian manifold and $E$ a real vector bundle over $\mathcal{M}$, which we equip with a non-degenerate metric $\langle\cdot,\cdot\rangle_{E}\in\Gamma^{\infty}(E^{\ast}\otimes_{s} E^{\ast})$. This induces a bilinear form $\langle\cdot,\cdot\rangle_{\Gamma^{\infty}(E)}$ on the level of sections defined by
$$\langle s,t\rangle_{\Gamma^{\infty}(E)}:=\int_{\mathcal{M}}\langle s,t\rangle_{E}\,\mathrm{vol}_{g}$$
defined for all $s,t\in\Gamma^{\infty}(E)$ with compactly overlapping support, where $\langle s,t\rangle_{E}\in C^{\infty}(\mathcal{M})$ is understood as $\langle s,t\rangle_{E}(p):=\langle s(p),t(p)\rangle_{E_{p}}$. In the theory of PDEs on manifolds, one often sees the following definition.
Consider a linear operator $D:\Gamma^{\infty}(E)\to\Gamma^{\infty}(E)$. Then, it is stated that there exists a unique linear operator $D^{\ast}:\Gamma^{\infty}(E)\to\Gamma^{\infty}(E)$, the "formal adjoint of $D$", such that
$$\langle Ds,t\rangle_{\Gamma^{\infty}(E)}=\langle s,D^{\ast}t\rangle_{\Gamma^{\infty}(E)}$$
for all $s,t\in\Gamma^{\infty}(E)$ with $\mathrm{supp}(s)\cap\mathrm{supp}(t)$ compact. Now, my question is:
Question: Why does this operator exists and why is it unique?
In general, $(\Gamma^{\infty}(E),\langle\cdot,\cdot\rangle_{\Gamma^{\infty}(E)})$ is not a Hilbert space, so we cannot use the standard 'Riesz representation theorem'-argument usually used in functional analysis for the proper definition of adjoint operators. Furthermore, I am not even sure that $\langle\cdot,\cdot\rangle_{\Gamma^{\infty}(E)}$ is non-degenerate, even if $\langle\cdot,\cdot\rangle_{E}$ is non-degenerate, so also uniquness is not clear to me.
Every comment and idea is welcome.
Here's a proof for a compact Riemannian manifold (I'm not very familiar with pseudo-Riemannian manifolds). Let $r$ be the rank of $E$, and let $m$ be the order of $D$.
Uniqueness follows by the standard argument from linear algebra: if we have operators $P, Q \colon \Gamma^\infty(E) \to \Gamma^\infty(E)$ such that $\langle Ds, t \rangle_{\Gamma^\infty(E)} = \langle s, Pt \rangle_{\Gamma^\infty(E)} = \langle s, Qt \rangle_{\Gamma^\infty(E)}$ for all $s, t$, then $0 = \langle s, (P-Q)t \rangle_{\Gamma^\infty(E)}$ for all $s, t$, so taking $s = (P-Q)t$ gives $P = Q$.
For existence, choose a finite cover of $M$ by coordinate charts $x_\ell \colon U_\ell \cong \mathbb{R}^n$ for $\ell = 1, \ldots, N$ over each of which is defined a trivialisation $E \rvert_{U_\ell} \to U_\ell \times \mathbb{R}^r$ whose induced local frame is orthonormal. Further, choose a partition of unity $\{\chi_\ell\}_{\ell=1}^N$ subordinate to the cover $\{U_\ell\}_{\ell=1}^N$. If we write $D = \sum_{\lvert \alpha \rvert \leq m} A_\ell^\alpha \partial^{\lvert \alpha \rvert}/\partial x^\alpha$ on $U_\ell$ for each $\ell$, then \begin{align} \langle Ds, t \rangle_{\Gamma^\infty(E)} &= \sum_{\ell=1}^N \langle Ds, \chi_\ell t \rangle_{\Gamma^\infty(E)} \\ &= \sum_{\ell=1}^N \int_{U_\ell} \sum_{\lvert \alpha \rvert \leq m} \biggl( A^\alpha_\ell \frac{\partial^{\lvert \alpha \rvert} s}{\partial x^\alpha}\biggr)^{\!\intercal} \chi_\ell t \operatorname{vol}_{g_\ell} \\ &= \sum_{\ell=1}^N \int_{U_\ell} s^\intercal \frac{1}{\sqrt{\det g_\ell}} \sum_{\lvert \alpha \rvert \leq m} (-1)^{\lvert \alpha \rvert} \frac{\partial^{\lvert \alpha \rvert}}{\partial x^\alpha} \Bigl( \sqrt{\det g_\ell} (A^\alpha_\ell)^\intercal \chi_\ell t \Bigr) \operatorname{vol}_{g_\ell} \\ &= \sum_{\ell=1}^N \int_{U_\ell} s^\intercal D_\ell^* t \operatorname{vol}_{g_\ell} \end{align} where the third line follows by integration by parts (there are no boundary terms because $x_\ell(U_\ell) = \mathbb{R}^n$), and in the fourth line we have defined \begin{equation} D_\ell^* t = \frac{1}{\sqrt{\det g_\ell}} \sum_{\lvert \alpha \rvert \leq m} (-1)^{\lvert \alpha \rvert} \frac{\partial^{\lvert \alpha \rvert}}{\partial x^\alpha} \Bigl( \sqrt{\det g_\ell} (A^\alpha_\ell)^\intercal \chi_\ell t \Bigr). \end{equation} Since the right-hand side of the above is supported $U_\ell$, this formula actually defines a global operator $D_\ell^* \colon \Gamma^\infty(E) \to \Gamma^\infty(E)$; moreover, it is a partial differential operator of order $m$. Hence $D^* = \sum_{\ell=1}^N D_\ell^* \colon \Gamma^\infty(E) \to \Gamma^\infty(E)$ is a partial differential operator of order $m$, and we have \begin{equation} \langle Ds, t \rangle_{\Gamma^\infty(E)} = \sum_{\ell=1}^N \int_{U_\ell} u^\intercal (D_\ell^* v) \operatorname{vol}_{g_\ell} = \sum_{\ell=1}^N \langle s, D_\ell^* t \rangle_{\Gamma^\infty(E)} = \langle s, D^* t \rangle_{\Gamma^\infty(E)}, \end{equation} which is the desired property.