OpenAlex ·
Reflexivity of Grothendieck Spaces and Strongly Normal Sequences
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- OpenAlex
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- arXiv (Cornell University)
- DOI
- 10.48550/arxiv.2610.12787
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Abstract
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 characterized by the subsequence behavior of normal sequences: a Grothendieck space $E$ is reflexive if and only if every normal sequence in $E^{*}$ contains a strongly normal subsequence. If $E$ is nonreflexive and Grothendieck, then $E^{*}$ contains a normalized weakly null sequence none of whose subsequences is strongly normal; more strongly, it contains a closed hereditarily $\ell_{1}$ 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 $E/\bigcap_n\ker z_n$ 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 $(L_{\infty}(μ))^{*}$ for nonatomic probability spaces (and, more generally, for $σ$-finite spaces with a nonatomic part) and in $\ell_{\infty}^{*}$.
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