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I would like to know,
Why is it true that $e_n$, the nth unit vector in $\ell^p(\mathbb N)$ converges weakly to $0$. $1 < p < \infty$ According to Mazur's lemma, which says that $y_n$ is convex combination of $e_n$ converges to $0$ in norm topology. I am trying to construct an explicit convex combination but i failed.

How do i construct it, any example would be nice. Thanks.

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    @David Mitra : Thank you :)2012-12-13

1 Answers 1

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I suppose here $1.

Note that by definition, $e_n\overset{w}{\to}0$ iff $f(e_k)\to 0, \forall f\in (l^p)'=l^q$, so for any $f=(a_k)\in l^q$, we check that $|f(e_k)|=|a_k|\to 0.$

For the second part, just take $y_n=\sum_{k=n+1}^{2n}\frac{1}{n}e_k$, then calculation shows that $||y_n||^p=\sum_{k=n+1}^{2n}\frac{1}{n^p}=\frac{1}{n^{p-1}}\to 0$, since $p>1$.

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    @Theorem Yes, you are correct.2012-12-13