Let $H$ be a Hilbert Space over a field $\mathbb{F}$ and $(e_n)_{n\in\mathbb{N}}$ be an orthonormal sequence in $H$.
Show that for any $x\in H$ if $$x=\sum_{k=1}^{n=\infty} a_k e_k $$ with $a_k\in\mathbb{F}$ $\implies a_k = \langle x,e_k\rangle$.
I have tried to use Bessel's inequality and Pythagoras' Theorem however I have not had any success.
Any help would be greatly appreciated.
This is just a calculation:
$$\langle x, e_k\rangle = \sum_{l=1}^\infty a_l \langle e_l, e_k \rangle = a_k$$
The sum could be put outside the inner product since the inner product is continuous.