Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question
up vote 1 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.