Recall that a Banach space $X$ is said to have the Dieudonné property if for every Banach space $Y$, an operator $T:X\rightarrow Y$ that transforms weakly Cauchy sequences into weakly convergent sequences is weakly compact. We denote by $H(X)$ the subset of $X^{**}$ formed by all the $\sigma(X^{**},X^{*})$-limits of weakly Cauchy sequences in $X$. A. Grothendieck (Canad J. Math, 1953) characterized the Dieudonné property as follows:
Let $X$ be a separable Banach space. The following statements are equivalent:
(1) $X$ has the Dieudonné property;
(2) Every $\sigma(X^{*},H(X))$-convergent sequence in $X^{*}$ is weakly convergent.
Question: Is there a self-contained or simple proof of this result?
