Reflexivity of Grothendieck Spaces and Strongly Normal Sequences

A normalized sequence $(y_n)$ in the dual $E^{*}$ of a Banach space $E$ is called normal if it is weak-star null, and strongly normal if [ G(y_n)=\left{x\in E:\sum_{n=1}^{\infty}|y_n(x)|<\infty\right} ] is dense in $E$. We prove that, within the class of Grothendieck Banach spaces, reflexivity is ch