Consider the space $E_n = C^1([0,1]^n)$ of continuously differentiable functions with the usual norm $$\max\{ \|f\|_\infty, \|f^\prime_{x_1}\|_\infty, \ldots, \|f^\prime_{x_n}\|_\infty\}.$$ making it a Banach space. (The norm with the subscript $\infty$ stands for the supremum norm.)
Inspired by the recent question:
Jean Bourgain's Relatively Lesser Known Significant Contributions
let me point out that Bourgain proved in 1983 that the dual of $E_n$ is weakly sequentially complete for any $n$.
Moreover, it is an exercise on Taylor's theorem that $E_1$ is isomorphic to $C[0,1]$ in which case the dual of $E_1$ is an abstract $L_1$-space so it is weakly sequentially complete and $E_1^{**}$ a Grothendieck space.
Is $E_n^{**}$ a Grothendieck space for every $n$?
Note that this, if true, would imply Bourgain's result as duals of Grothendieck spaces are weakly sequentially complete so here even the tridual of $E_n$ would have this property.
Tracking citations of Bourgain's paper gives evidence that this is an open problem but it is not inconceivable that it follows from a combination of results proved later.
