Let $T:X\to X^*$ where $X$ is some Banach space.
Under what condition can we guarantee that $T$ satisfies the following:
$$\langle x_n,Ty_n\rangle\to \langle x,Ty\rangle$$
whenever $x_n$ converges weakly to $x$ and $y_n$ converges weakly to $y$?
Are there any known result what properties of $X$ are required such that this holds for the normalized duality map $J$ of $X$?