Definition: Let $\ell_2$ be the linear space that consists of all sequences $x=(x_n)$ in $\mathbb{F}$ for which $\sum\limits_{n=1}^{\infty}|x_n|^2<\infty$. Then $\|x\|_2=(\sum\limits_{n=1}^{\infty}|x_n|^2)^{\frac{1}{2}}$ is a norm on $\ell_2$, and $\ell_2$ is a Banach space with respect to this norm. Let $c_{00}$ be the linear space of all sequences in $\mathbb{F}$ that are eventually zero. Then $c_{00}$ is a linear subspace of $\ell_2$.
Equip $c_{00}$ with $\|\cdot\|_2$. Let $B$ be the norm closed unit ball of $c_{00}$. I want to show $B$ is not weakly compact. It suffices to show that there every sequence $(x_n)$ in $B$ such that has no weakly cluster point $x$ in $B$.
In fact, let $y=(y_n)\in\ell_2$. Then $f_y(x)=\sum\limits_{n=1}^{\infty}x_ny_n$ ($x=(x_n)\in c_{00}$) defines a bounded linear functional on $c_{00}$; that is $f_y\in {c_{00}}^*$.
I saw this. And I am trying to use it to find an example. But I am stuck. Can anyone help me? Thank you!