I have to prove that if the series $$\sum x_n$$ converges weakly then it converges (normally)
my solution goes as follows:
$\langle \sum x_n | y\rangle \rightarrow \langle x|y\rangle $, for all $y\in H$ (Hilbert space).
Then, $\sum \langle x_n| \rightarrow \langle x| \iff \langle \sum x_n|\rightarrow \langle x| \iff \sum x_n \rightarrow x$
the thing is the exercises says that $x_n$ are orthonormal, which I didn't use.