Let $E$ be a $\mathbb R$-Banach space, $\Theta\subseteq C^0(E,E)$ be a $\mathbb R$-Banach space and $G\in\Theta'$.
The "support" $\operatorname{supp}G$ of $G$ is the set of all $x\in E$ such that for every open neighborhood $U$ of $x$, there is a $\theta\in\Theta$ with $\operatorname{supp}\theta\subseteq U$ and $\langle G,\theta\rangle\ne0$.
Now, if $\Omega\subseteq E$ and $\langle G,\theta\rangle=0$ for all $\theta\in\Theta$ with $\left.\theta\right|_\Omega=0$, can we infer that $\operatorname{supp}G\subseteq\Omega$?
Take $\theta\in\Theta$ such that $\operatorname{supp}\theta \subseteq \complement\Omega.$ Then $\theta|_\Omega=0$ so $\langle G, \theta \rangle=0.$ This means that $\operatorname{supp}G \subseteq \complement(\complement\Omega)=\Omega$ according to the definition of support that I have learnt: $$\operatorname{supp}G=\complement \bigcup \{ \text{open } U\subseteq E \mid \langle G, \theta \rangle=0 \text{ for all $\theta\in\Theta$ such that $\operatorname{supp}\theta \subset U$} \},$$ which I think is equivalent to your definition $$ \operatorname{supp}G=\{x\in E \mid \forall \text{ open } U\ni x \, \exists \theta\in\Theta \text{ with } \operatorname{supp}\theta\subset U \text{ s.t. } \langle G, \theta \rangle \neq 0\}. $$