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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 $H^0(\Omega),\cdots,H^k_0(\Omega)$?

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hint Look up the spectral theorem for self-adjoint operators. Apply it to the Laplacian on your domain $\Omega$. (The spectral theory of the Laplacian is rather standard, and should be done in most text books on PDEs or functional analysis; see, e.g. Evans, Partial Differential Equations). Now notice that on $H^1_0(\Omega)$ you have the following integration by parts identity:

$$ \int_{\Omega} \nabla u \cdot \nabla v ~dx = - \int_{\Omega} u \triangle v ~dx $$

where $\triangle$ is the Laplace operator.

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