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Can anyone provide an example of a sequence $(x_{k}) \in L^p(\mathbb{N})$ with $1<p<\infty$ such that $x_{k}(n)\rightarrow 0$ as $k\rightarrow \infty$ such $x_{k}$ doesn't converges weakly to zero.

The sequence has to be unbounded because if the sequence is bounded with this the sequence is weakly convergent.

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Weakly convergent sequences are norm bounded, so any unbounded sequence will work.

For an explicit example in $\ell_2$, take $x_n=ne_n$, where $e_n$ is 0 in all coordinates save the $n$'th, which is 1. Then the functional $x^*=(1,1/2,1/3,\ldots)$ verifies that $(x_n)$ is not weakly convergent to the zero vector (and thus not weakly convergent).

Similar examples can be constructed for other $\ell_p$ spaces.

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  • $\begingroup$ The same example holds for $\ell_{p}$. Including the same functional. Thanks for clearing it up! $\endgroup$ – checkmath Feb 25 '12 at 2:52

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