Function restriction of a distribution

Question

Let $V, \Omega$ open subsets of $\mathbb{R^d}$, with $V \subset \Omega$. The function restriction $R:\mathcal{D}'(\Omega) \rightarrow \mathcal{D}'(V)$ given by $u \mapsto u|_{V}$ is continuous. If we consider $S:\mathcal{E}'(\Omega)\cap \{u \in \mathcal{E}'(\Omega): \hbox{ supp } u \subset V \} \rightarrow \mathcal{E}'(V)$, given by $Su=u|_V$. Is the function $S$ injective?