A sequence $(x_n)$ in a Banach space $X$ is called weakly Cauchy if for every $\ell \in X'$ the sequence $(\ell(x_n))$ is Cauchy in the scalar field.
I want to show that weakly Cauchy sequences are not necessarily weakly convergent.
This seems to be the case for Hilbert spaces On the limits of weakly convergent subsequences, whats the difference?