# Is $H^1_0(\Omega)$ dense in $L^2(\Omega)$?

Is $H^1_0(\Omega)$ dense in $L^2(\Omega)$ for bounded domains? It is true for $H^1$ functions of course but what about this subset?

Sorry for the elementary question but I never see this so I think the answer is it's not.

• $C_0^\infty(\Omega) \subset H_0^1(\Omega) \subset L^2(\Omega)$. – Siminore Apr 12 '13 at 10:24
• as $C^\infty_0$ is dense in $L^2$, so is $H^1_0$. How stupid of me. Thanks – michael_faber Apr 12 '13 at 10:46

1) $\overline{C_0^{\infty}(\Omega)}^{||\cdot||_{2}} = L^2(\Omega)$
2) $\overline{C_0^{\infty}(\Omega)}^{||\cdot||_{H^1}} = H_0^1(\Omega)$
3) $C_0^{\infty} \subset H^1_0(\Omega) \subset L^2(\Omega)$ and taking the closure with respect the $L^2$-norm we get $L^2(\Omega) \subset \overline{H^1_0(\Omega)}^{||\cdot||_2} \subset L^2(\Omega)$.