A non weakly convergent sequence in $L^p(\mathbb{N})$

Question

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.

Answer 1

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.