Let $\Omega, \Omega'$ be two open subsets of $\mathbb{R}^{n}$ such that $\Omega \subset \Omega'$. Let $f \in C_c^\infty(\Omega)$ and $\tilde{f}$ be the trivial extension of $f$ to $\Omega'$, that is, $\tilde{f}=f$ in $\Omega$ and $\tilde{f}=0$ in $\Omega'\setminus \Omega.$ Let $J:C_c^\infty(\Omega) \to C_c^\infty(\Omega')$ given by $J(f)=\tilde{f}$. The transpose of $J$, ${}^tJ:\mathcal{D}'(\Omega') \to \mathcal{D}'(\Omega)$, is called the restriction to $\Omega$ of distributions in $\Omega'$.
In Trèves book he show that $${}^tJ:\mathcal{D}'(\Omega') \to \mathcal{D}'(\Omega)$$ is not one-to-one and states:
(2') the restriction mapping from $\mathcal{D}'(\Omega')$ to $\mathcal{D}'(\Omega)$ is not onto.
The proof of (2') is more complicated, and will not be given, but the student should keep in mind the two facts above.
My question: I would like to know how to prove that ${}^tJ$ is not onto.