Let $\Omega \in R^n$ be open, bounded and with smooth boundary. Can you prove the existence of a system of vectors that simultaneously forms an orthogonal basis both in $L^2(\Omega)$ and $H^1_0(\Omega)$?
Can you generalize this construction to obtain a simultaneous orthogonal basis for $H^0(\Omega),\cdots,H^k_0(\Omega)$?