Reflexivity of Grothendieck Spaces and Strongly Normal Sequences

@misc{sliwa2026reflexivity,
  title = {{Reflexivity of Grothendieck Spaces and Strongly Normal Sequences}},
  author = {Wiesław Śliwa},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12787},
  eprint = {2610.12787},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12787},
  note = {CC BY-NC-ND 4.0},
}

TY  - UNPB
TI  - Reflexivity of Grothendieck Spaces and Strongly Normal Sequences
AU  - Wiesław Śliwa
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.12787
UR  - https://arxiv.org/abs/2610.12787
AB  - 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 ℓ_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/⋂_nker 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_∞(μ))^* for nonatomic probability spaces (and, more generally, for σ-finite spaces with a nonatomic part) and in ℓ_∞^*.
ER  -