Let $\Omega$ be a bounded subset of $\mathbb{R}^d$ with smooth boundary. It is shown in Taylor's Partial Differential Equations I, for instance, that the dual of $H^1(\Omega)$ is isomorphic to the subspace of $H^{-1}(\mathbb{R}^d)$ consisting of distributions supported on $\bar{\Omega}$.
I wanted to check if $C_c^{\infty}(\Omega)$ is dense in this Banach space. Exercise 4.5.13 of Taylor seems to suggest the affirmative, but I am not sure if the problem refers to all $s\in \mathbb{R}$ or just $s\geq 0$.