I know that if $H$ is a Hilbert space and $(e_{j})_{j\in\mathbb{N}}$ is an orthonormal system in $H$ and $f\in H$. Then one has Bessel's inequality $$\sum_{j=1}^{\infty}|\langle f,e_{j}\rangle |^{2}\leq||f||^{2}<\infty.$$
If I want a similar statement for $\langle e_{j},f\rangle $ is it true that:
$$\sum_{j=1}^{\infty} |\langle e_{j},f\rangle |^{2} = \sum_{j=1}^{\infty}|\langle \overline{f,e_{j}}\rangle |^{2}\leq||f||^{2}<\infty.$$
Or am I doing something that is not allowed?
How do I know that $(\overline{e_{j}})_{j\in \mathbb{N}}$ is an orthonormal system in $H$?
