Let $V \subset H$ be a pair of Hilbert spaces (with different inner products). The embedding is continuous and dense, and both spaces are separable.
Is it always the case that one can I find a sequence $\{v_n\}$ such that $v_n$ are an orthogonal basis in $V$ and they are also an orthonormal basis in $H$? (Eg. if $V=H^1_0$ and $H=L^2$ then the eigenfunctions of the Laplacian can be $v_n$).
If $V \subset H$ is a compact embedding, I think this is related to Hilbert-Schmidt theory. But what operator do I use to apply the theory?