Reflexivity of Grothendieck Spaces and Strongly Normal Sequences

A normalized sequence $(y_n)$ in the dual $E^{\ast }$ 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 characterized by the subsequence behavior of normal sequences: a Grothendieck space $E$ is reflexive if and only if every normal sequence in $E^{\ast }$ contains a strongly normal subsequence. If $E$ is nonreflexive and Grothendieck, then $E^{\ast }$ contains a normalized weakly null sequence none of whose subsequences is strongly normal; more strongly, it contains a closed hereditarily ${\mathrm{\ell }}_1$ subspace $Y$ with a normalized weakly null Schauder basis such that no sequence in $S(Y)$ is strongly normal. We also obtain an exact quotient criterion valid for arbitrary Banach spaces: a normal sequence has a strongly normal subsequence precisely when, for some subsequence $z=(z_n)$, the quotient $E\mathrm{/}{\bigcap }_n\ker z_n$ is separable. The proofs combine a weak-star basic-sequence argument with results of Haydon, Johnson–Maurey–Schechtman and Popov. Concrete realizations are given in $(L_{\mathrm{\infty }}(\mu ){)}^{\ast }$ for nonatomic probability spaces (and, more generally, for $\sigma$-finite spaces with a nonatomic part) and in ${\mathrm{\ell }}_{\mathrm{\infty }}^{\ast }$.