I'm concluding something that is not convincing me.
Imagine a sequence of elements $x_k$ in an Hilbert space $\mathcal{H}$, then if it is weakly convergent to $x$ that means $$ \lim_{k\to +\infty} \langle x_k - x, y\rangle = 0 \,\forall y\in\mathcal{H} $$ In particular this holds for all the elements $y=e_j\in\mathcal{H}$ where $\{e_j\}$ constitutes a complete orthonormal sequence in a separable Hilbert space, so we have $$ \lim_{k\to +\infty} \langle x_k - x, e_j\rangle = \left\langle \lim_{k\to +\infty} x_k - x, e_j\right\rangle = 0 \,\forall j $$ but Hilbert space is indeed separable so we have $$ \lim_{k\to +\infty} x_k - x = \sum\limits_j \left\langle \lim_{k\to +\infty} x_k - x, e_j\right\rangle e_j = 0 $$ (see for example "Hilbert spaces with applications - Debnath, Mikusinski" third edition pag. 115 theorem 3.4.14) and this means $$ \left\lVert \lim_{k\to +\infty} x_k - x\right\rVert = \lim_{k\to +\infty} \lVert x_k - x\rVert = 0 $$ so the sequence $\{x_k\}$ is strongly convergent to its weak limit. What am I doing wrong?