Let $X$ be a Banach space.
For $f\in\mathcal S'(\mathbb R^n,X)$ and $G\subset_{\text{open}}\mathbb R^n$, we define $$f=0\ \mathrm{on}\ G\iff \langle f,\varphi\rangle=0\ \mathrm{for\ each}\ \varphi\in\mathcal S(\mathbb R^n,\mathbb C)\ \mathrm{with}\ \operatorname{supp} \varphi\subset G$$
Then, I want to find whether or not $f=0$ on $\underset{\text{$G$ open, $f=0$ on $G$}}\bigcup\quad\ \ G$ is true.
So, let $\varphi\in\mathcal S(\mathbb R^n,\mathbb C)\ \mathrm{with}\ \operatorname{supp} \varphi\subset\underset{\text{$G$ open, $f=0$ on $G$}}\bigcup\quad\ \ G.$ If I can say $\langle f,\varphi\rangle=0$, it will follow that $f=0$ on $\underset{\text{$G$ open, $f=0$ on $G$}}\bigcup\quad\ \ G$. But I cannot find the reason why $\langle f,\varphi\rangle=0$.
Is that the true statement ?