Suppose that $\langle X,Y\rangle$ is a dual pair of Banach spaces satisfying $|\langle x,y\rangle|\leq \Vert x\Vert\Vert y\Vert$ for all $x\in X$, $y\in Y$. 


Is it true that the unit ball of $X$ is $\sigma(X,Y)$-closed?



weak closedness of the unit ball for a dual pair of Banach space
Okay, even assuming that $Y$ separates points the answer is still no. Take $X = l^1$ and let $Y$ be the set of elements $(a\_n)$ of $l^\infty$ which satisfy $\lim a\_n = 2a\_1$. It's easy to see that $Y$ separates the points of $X$, but $e\_n \to 2e\_1$ weakly where $(e\_n)$ is the standard basis of $l^1$. So $2e\_1$ is in the weak closure of the unit ball.


