Let $H$ be a Hilbert space with orthonormal basis $(e_i)_{i\in I}$, where $I$ is an uncountable index set. How to prove that for any $x\in H$, there exists countable $(e_i)_{i\geq 1}$ such that $$x=\sum_{i=1}^{\infty}\langle x,e_i\rangle e_i.$$
Idea: Since $(e_i)_{i\in I}$ is an orthonormal basis, then $\mathrm{span}((e_i)_{i\in I})$ is dense in $H$, so there exists a sequence $(x^{(n)})_{n\geq 1}$ which converges to $x$, and for each $x^{(n)}$, it can be represented as a finite linear combination: $$x^{(n)}=\sum_{i\in J_n}x^{(n)}_i e_i,$$ where $J_n\subset I$ is finite index set.
Another question: We know that if Hilbert space $H$ has a countable orthonormal basis $(e_i)_{i\geq 1}$, then $\forall x\in H$, we have the Parseval identity $x=\sum_{i=1}^{\infty}\langle x,e_i\rangle e_i$. But in this case where the orthonormal basis is uncountable, does the Parseval identity $x=\sum_{i\in I}\langle x,e_i\rangle e_i$ hold?
