Let $1\le p< \infty$. Show that a sequence $t_k = ({t_{kj}})_{j=1}^{\infty}\in l^p$ converges weakly to 0 iff $||t_k||_p$ is bounded and $\lim_k t_{kj}=0$. I proved that if $t_k$ converges weakly to 0 then we conclude that. I want to prove the reciprocal.
Let's assume that $1<p<\infty$ If I assume that $(t_k)$ it's weakly cauchy I can prove that it's weakly convergento to 0, but I don't know how to prove that. Under that assumption I used the reflexivity of the space $l^p$ ($l^1$ is not reflexive)
If $p=1$ since $l^1$ is not reflexive my arguments are not valid here. I don't really now if the result it's also true here. Please help me!
You can find a proof of more general result here.
Appply that theorem to the case $p\in(1,+\infty)$ with $S=\{f_j\in (\ell_p)^*:j\in\mathbb{N}\}$, where $$ f_j:\ell_p\to\mathbb{K}: t\mapsto t_j $$
For $p=1$ see this answer, where it was proved that weak convergence is equivalent to strong convergence. It is remains to note that every strongly convergent sequence is bounded and pointwise convergent.