Inner-product equality

Question

I am reading Halmos. For some Hilbert space $H$ he has shown $$ \sum_j |(e_i-f_i,f_j)|^2 = ||e_i-f_i||^2. $$Here $(e_n)$ is an orthonormal basis and $(f_n)$ is some orthnormal set. He directly concludes $\sum_j (e_i-f_i,f_j)f_j = e_i-f_i$. I do not see how this follows. I understand that $(e_i-f_i,f_j)f_j$ is the projection of $e_i-f_i$ onto $f_j$, although I don't see why we can have equality when $(f_n)$ is just an orthonormal set and not a basis.

Answer 1

This is the Pythagorean theorem. $$ \sum |\langle x, f_j\rangle|^2 + \| x-\sum \langle x, f_j\rangle f_j\|^2 = \|x\|^2. $$ So $x = \sum \langle x, f_j\rangle f_j$ if and only if $\sum |\langle x, f_j\rangle|^2 = \|x\|^2$.