Let ${v_n}_{n=1}^\infty$ be a closed orthonormal system in $L^2[0,1]$. We want to prove that for every $a\in[0,1]$, we have
$$a=\sum_{n=1}^\infty \left|\int_0^a \overline{v_n(x)}dx\right|^2$$
where $\overline{v_n(x)}$ denotes the complex conjugate of $v_n(x)$.
now the solution was like this , but I have no idea what is that and how they solved it using this function and someone explain please \begin{equation*} |1_{[0,a]}|2 = {\sum |\langle 1_{[0,a]}, \phi_n\rangle|^2} \end{equation*}