A complete orthonormal system $\{e_i\}^\infty_{i=1}$ in $H$ is a basis in $H$

Question

I'm studying about Hilbert space from a book of functional analysis and I just read this theorem (2.1.10) and its' proof.

I cannot understand why $(y-x)\perp e_i$? why is it implied?

Answer 1

In Hilbert spaces the dot product is closed to infinite sums as well if the series converges. So because the system is orthonormal:

$\langle y-x,e_j\rangle=\langle \sum_{i=1}^\infty\langle x,e_i\rangle e_i,e_j\rangle-\langle x,e_j\rangle=\sum_{i=1}^\infty\langle x,e_i\rangle\langle e_i,e_j\rangle-\langle x,e_j\rangle=$

$=\langle x,e_j\rangle-\langle x,e_j\rangle=0$