I'm stuck on the below question and was wondering if someone could maybe give me a hint? My idea is to show that $\operatorname{span}(\bigcup E_n)$ is dense in $D$, but if I'm completely honest I have no idea where to begin.
Let $H$ be a Hilbert space and $D \subset H$ be such that the linear space spanned by $D$ is dense in $H$. Let $(E_n)_n$ be a sequence of mutually orthogonal closed linear subspaces of $H$. Prove that if\ $$\sum_{n=1}^{\infty}\|P_{E_n}u\|^2 =\|u\|^2 \quad \forall u \in D$$ then $H = \bigoplus^{\infty}_{n=1}E_n$.
Many Thanks