I have a question, which maybe looks very simple!
It is known that if a sequence $\{f_{n}\}_{n\geq 1}$ is an orthonormal basis for a separable Hilbert space $H$ then $\sum_{n=1}^{\infty}|\langle f, f_{n}\rangle|^{2}=\|f\|^{2}$ for all $f\in H$.
Is the following correct:
Let $I,J$ be two disjoint subsets of the integers $\{1,2,3,...\}$ such that $I\cup J=\{1,2,3,...\}$, then
$\sum_{n\in I}|\langle f, f_{n}\rangle|^{2}+\sum_{n\in J}|\langle f, f_{n}\rangle|^{2}=\|f\|^{2}$ so we will have $\sum_{n\in I}|\langle f, f_{n}\rangle|^{2}=\|f\|^{2}-\sum_{n\in J}|\langle f, f_{n}\rangle|^{2}$ and both of the summands are finite, and we can say
$\sum_{n\in I}|\langle f, f_{n}\rangle|^{2}=A \;(\text{which depends on}\;f)$ $\sum_{n\in J}|\langle f, f_{n}\rangle|^{2}=B \; (\text{depends on}\;f)$ and $A+B=\|f\|^{2}$,
I'm just worried about convergence!