In the book I used to study they said that for any hilbertian basis $(e_n)_n$ we have that $e_n$ converges weakly to the null vector.
I did not succeed to prove it except in the particularly case of $\ell_2$.
Here is the attempt I tried for any separable Hilbert space $H$ :
$$ \lVert \langle e_n, v\rangle \rVert = \lVert \langle e_n, \sum_{i=1}^{\infty}\langle v, e_i\rangle e_i \rangle \rVert = \lVert \sum_{i=1}^{\infty}\langle v, e_i\rangle\langle e_n, e_i \rangle \rVert\leq \sum_{i=1}^{\infty}\lvert \langle v, e_i\rangle\lvert\lVert\langle e_n, e_i \rangle \rVert $$
And from there I don’t know how to pursue (if it is the good starting point).
Have you some hint to provide please ?
Thank you a lot