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.


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.

  • $\begingroup$ 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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