I have a problem with a line at the end of an example who tolds this: let $(E,||-||)$ be a separable Banach space. If there exist a weak Cauchy sequence who doesn't converge weakly to an element of $E$, then the weak topology on $E$ is not metrizable.
I can't undestand why necessairly the weak topology is not metrizable by the condition about the Cauchy sequence.
I think that must exist a result like that, who prove the above assertion, but I don't know if this result is valid:
If $ (E, ||-||) $ is a separable Banach space, then the metric space $ (E,d) $ is complete for all metric $ d $ whose induced topology is coarser than the topology induced by $ ||-|| $.
If this proposition is valid, (we know that the weak topology is the coarsest topology such that all linear functionals of the dual space are continuous,) so the first assertion will be proven.
PS: I do not want to use the fact that if $E$ is an infinite dimensional normed space, then the weak topology is not metrizable.