For a tempered distribution $f\in\mathcal S'(\mathbb R^n,X)$ and an open set $O\subset\mathbb R^n$,
$f=0$ on $O$ $\underset{def}\iff$ for each $\varphi\in \mathcal S(\mathbb R^n)$ with supp$\varphi\subset O$, $\langle f,\varphi\rangle=0.$ (supp$\varphi=\overline{\{x\in\mathbb R^n\mid \varphi(x)\neq 0\}}.$), and supp$f$ is defined as the compliment of the maximal open set on which $f$ satisfied $f=0.$
Then, I'm confused with this statement.
Let $f\in\mathcal S'(\mathbb R^n,X)$ and $x\in\mathbb R^n$. Then, $x\notin supp f$ if and only if there is a neighbourhood of $x$, say $U$, such that $\langle f ,\varphi\rangle=0$ for all $\varphi\in C_0^\infty(U).$
I don't know what the last part : $\langle f ,\varphi\rangle=0$ for all $\varphi\in C_0^\infty(U)$ means.
In the context, $f$ is a function with domain $\mathcal S(\mathbb R^n)$. So in order to write $\langle f,\varphi\rangle$, $\varphi$ has to belong to $\mathcal S(\mathbb R^n)$. But $\varphi\in C_0^\infty(U)$ is a function defined not on $\mathbb R^n$ but on $U$. So I think $\varphi$ doesn't belong to $\mathcal S(\mathbb R^n)$. and hence $\langle f,\varphi\rangle$ isn't defined.
How should I interpret $\langle f ,\varphi\rangle$ for $\varphi\in C_0^\infty(U)$ ?
In my one idea, I consider $\varphi\in C_0^\infty(U)$ as $\varphi\in C_0^\infty(\mathbb R^n)$ with $\varphi(x)=0$ for $x\notin U.$ (I'm not sure this interpretation is correct.)