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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!

Davide Giraudo
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Shanks
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1 Answers1

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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.

José Carlos Santos
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Norbert
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  • Thanks Norbert for the complete answer! But I have one question! why the span of the projections are dense? – Shanks Oct 17 '13 at 05:14
  • span of projections is the space of finitely supported vectors usually denoted by $c_{00}$. It is dense in $\ell_p$ by the following argument. For a given $\varepsilon$ and $t\in\ell_p$ there exists $N\in\mathbb{N}$ such that $\sum_{j=N+1}^\infty|t_j|^p<\varepsilon^p$. Then consider $y\in c_{00}$ such that $y_j=t_j$ for $j=1,\ldots,N$ and $y_j=0$ otherwise. Then $\Vert t- y\Vert_p<\varepsilon$. – Norbert Oct 17 '13 at 06:19
  • You are right, thanks! – Shanks Oct 17 '13 at 06:32
  • But note that for the case $p=1$ the dual of $l^1$ is $l^{\infty}$ and here $c_{00}$ is not dense on $l^{\infty}$ so the span of the projections are not dense on $(l^1)'$ – Shanks Oct 17 '13 at 07:26
  • @Shanks, you are right. See edits to my answer. – Norbert Oct 17 '13 at 09:06