Uniqueness of a "restriction" of a local operator $\alpha:\Gamma(E)\to \Gamma(F)$, where $E,F\to M$ are smooth vector bundles Let $E$ and $F$ be smooth vector bundles over the same manifold $M$. Recall that a local operator $\alpha:\Gamma(E)\to \Gamma(F)$ (where $\Gamma(E)$ is the $\Bbb R$-vector space of all smooth sections $M\to E$) is an $\Bbb R$-linear map such that if $s|_U=0$ for some $s\in \Gamma(E)$ and open $U\subset M$, then $\alpha(s)|_U=0$.
I am trying to prove the following statement: Given such a local operator $\alpha$, then for each open subset $U\subset M$, there is a unique $\Bbb R$-linear map $\alpha|_U:\Gamma(U,E)\to \Gamma(U,F)$ ($\Gamma(U,E)$ is the space of all smooth sections $U\to E$) such that $\alpha|_U(s|_U)=\alpha(s)|_U$ for all $s\in \Gamma(E)$. I've shown existence, but how can we prove uniqueness?
 A: Here's one argument using bump functions.
Let $\beta,\gamma:\Gamma(U,E)\to\Gamma(U,F)$ be two such restrictions. It suffices to show that $\beta-\gamma=0$.
Choose any local section $s\in\Gamma(U,E)$, and any $x\in U$. It now suffices to show that $(\beta-\gamma)s(x)=0$.
We may always find a compactly supported bump function $\psi:M\to\mathbb{R}$ such that $\psi=1$ on an open neighborhood $V\ni x$ and $\text{supp}(\psi)\subset U$. We know by locality that $(\beta-\gamma)s(x)=(\beta-\gamma)(\psi s)(x)$ (since $s-\psi s$ vanishes on $V$), and since $\psi s$ may be smoothly extended to a section $\widetilde{\psi s}$ on all of $M$, we have
$$
(\beta-\gamma)s(x)=(\beta-\gamma)(\psi s)(x)=\beta(\psi s)(x)-\gamma(\psi s)(x)=\alpha(\widetilde{\psi s})(x)-\alpha(\widetilde{\psi s})(x)=0
$$
Edit:
As probably123 points out, this proof shows that the restriction $\alpha|_U$ is unique among local operators. If there is no requirement that the restriction be local, then there is no guarantee of uniqueness.
The space of restricted sections $\Gamma(M,E)|_{U}:=\{s|_U:s\in\Gamma(M,E)\}$ is a proper subspace of $\Gamma(U,E)$ whenever there exist local  sections that cannot be smoothly extended. This means that the quotient $Q:=\Gamma(U,E)/\Gamma(M,E)|_U$, as well as its algebraic dual $Q^*$, are nontrivial.
In this case, we may choose a nonvanishing section $t\in\Gamma(U,F)$ and a nonzero element $\lambda\in Q^*$. Define a local operator $\psi:\Gamma(U,E)\to\Gamma(U,F)$ by $\psi(s)=\lambda([s])t$ (where $[\ ]$ denotes the projection into $Q$). Since $\psi$ vanishes on $\Gamma(M,E)|_U$, for any restriction $\alpha|_U$, $\alpha|_U+\psi$ is also a valid and distinct restriction.
A more explicit counterexample is hard to find in the smooth category since $Q^*$ is rather difficult to describe. It may also be possible to ensure uniqueness in other ways, such as topologizing the spaces of sections and requiring continuity, or restricting attention to compactly supported sections.
