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.
$x\in\ell_q$
. Note its coordinates converge to $0$. Then, what must $x(e_i)$ converge to? For the convex combinations, consider $y_n={1\over n}\sum_{j=1}^n e_j$. – 2012-12-13$p=1$
, $(e_i)$ is not weakly convergent and any convex combination of them will have norm 1. I'm not sure what happens when $p=\infty$. The $(y_n)$ I defined previously do tend to $0$ in norm. I'm not sure if $(e_i)$ converges weakly or not. – 2012-12-13