I'm going through a proof from Evans, and I'm stuck on what I think must be an easy linear algebra question.
Let $\{w_k\}_{k=1}^\infty$ be an orthonormal basis of $L^2(U)$, where $w_k\in H_0^1(U)$ is an eigenfunction with eigenvector $\lambda_k$.
If $u\in H_0^1(U)$, we can write
$u = \sum_{k=1}^\infty d_k w_k$
where $d_k = (u,w_k)_{L^2(U)}$. Additionally,
$\sum_{k=1}^\infty d_k^2 = \|u\|^2_{L^2(U)}$.
Why does this last equality hold? Is there some simple way to determine $||u||^2_2$ in terms of the orthonormal basis?
Any help appreciated!