$L^2(U)$ compact embedded in $H^{-1}(U)$?

Question

Let $U$ be an open subset of $R^d$. We already knew that $L^2(U)$ is a subset of $H^{-1}(U)$. Question: is this a compact embedding?

Answer 1

Yes, this embedding is compact. This can be seen, since this is the adjoint of the embedding of $H_0^1(U)$ in $L^2(U)$.