We know that spherical harmonics are a complete orthonormal system for $L^2(\mathbb{S}^2)$. Is it true that they are also a complete orthonormal system for $H^1(\mathbb{S}^2)$?
Furthermore, is it true that the eigenfunctions of the Laplacian on a domain $\Omega \subset \mathbb{R}^n$ are a complete orthonormal system?
I expect the answer to be no, I did some research on the internet and could find no answer. Any help would be greatly appreciated.