Let me first introduce some definitions. Let $1\leq p\leq \infty$.
A sequence $(x_{n})_{n}$ in a Banach space $X$ is said to be weakly $p$-convergent to $x\in X$ if the sequence $(x_{n}-x)_{n}$ is weakly $p$-summable in $X$.
We say that a sequence $(x_{n})_{n}$ in a Banach space $X$ is weakly $p$-Cauchy if for each pair of strictly increasing sequences $(k_{n})_{n}$ and $(j_{n})_{n}$ of positive integers, the sequence $(x_{k_{n}}-x_{j_{n}})_{n}$ is weakly $p$-summable in $X$.
We say that a Banach space $X$ is weakly sequentially complete of order $p$ if every weakly $p$-Cauchy sequence in $X$ is weakly $p$-convergent. The weakly sequential completeness of order $\infty$ is precisely the classical weakly sequential completeness.
It is known that neither the James space $J$ nor any of its higher duals is weakly sequentially complete.
Question: Is the dual $J^{*}$ of the James space $J$ weakly sequentially complete of order 2?
Thank you!
