I want to find a uniformly bounded sequence $\{x_n\}$ in $l^2(\mathbb{C})$ such that
$x_n$ does not converge to zero in weak topology, i.e., $\exists ~y\in l^2(\mathbb{C}),$ such that $\langle y, x_n\rangle\not\to 0$,
but $\{x_n\}$ satisfies the following condition:
$\lim_m\lim_n\langle x_{n+m},x_n\rangle=0$ or the stronger condition:
$\lim_n\langle x_{n+m},x_n\rangle=0, \forall m\geq 1.$
Thanks in advance!
Remarks:
1, Jacob Schlather has solved it for the case $\{x_n\}$ is not uniformly bounded, I have added the assumption that $\{x_n\}$ is uniformly bounded, which I forgot to add before.
2, This is one ''remark'' in page 85 of the book-- H.Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton Univ. Press, unless I misunderstand the meaning in the book.
It says: "It should be noted that the analogous result for ordinary convergence does not hold".
Lemma 4.9. Let $\{x_n\}$ be a bounded sequence of vectors in Hilbert space and suppose that $D-\lim_m(D-\lim_n\langle x_{n+m}, x_n\rangle)=0$ Then with respect to the weak topology, $D-\lim_nx_n=0$