I'm trying to find a reference to show the approximation of $H^{k}(\Omega)$ by $C^{\infty}(\Omega)$, where $\Omega$ is equal to $R^{n}$ for $n \geq 1$.
The only things I found are the global approximation theorems by smooth functions, e.g. Evans 5.3.2 and 5.3.3. Is the statement not true or did I fail to find the right book?
Compactly supported smooth functions ARE dense in $H^k(\mathbb{R}^n)$ (if I'm not mistaken..). Because the integrability implies that they vanish at infinity. On a bounded domain, $H^k$ doesn't necessarilly vanish at the boundary. As seen here https://en.wikipedia.org/wiki/Sobolev_space#Approximation_by_smooth_functions, you're good to go if you don't require compact support.