Proposition 10

If, in a hilbertian space E, a filter $\mathfrak{F}$ converges weakly to $x_0$, and if moreover $\lim_{\mathfrak{F}} \|x\| = \|x_0\|$, then $\mathfrak{F}$ converges to $x_0$ for the initial topology of E.

In fact, $\|x - x_0\|^2 = \|x\|^2 - 2\Re \langle x|x_0 \rangle + \|x_0\|^2$. Since $\langle x|x_0 \rangle$ tends to $\|x_0\|^2$ with respect to $\mathfrak{F}$ by hypothesis, and $\|x\|$ tends to $\|x_0\|$ with respect to $\mathfrak{F}$, $\|x - x_0\|$ tends to 0 with respect to $\mathfrak{F}$, hence the proposition.