PreprintCC BY 4.0arXiv
Normal sequences without strongly normal subsequences
Abstract
A sequence (yn*) in the unit sphere of E* is strongly normal if the vectors x∈ E satisfying ∑n|yn*(x)|<∞ form a dense subspace. For every set Γ of cardinality at least the continuum, we construct a normalized weakly null sequence in ℓ1(Γ)* with no strongly normal subsequence. This answers a question of Śliwa negatively in ZFC. Together with the classical selection argument below the bounding number b, the construction gives positive and negative bounds for this subsequence property on ℓ1(κ). Its validity on ℓ1(ω1) is independent of ZFC. The intermediate range bleqslantκ<c remains undecided by these results. We also give a counterexample in a Banach sequence space that is not isomorphic to any ℓ1(Γ).
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BibTeX
@misc{kakol2026normal,
title = {{Normal sequences without strongly normal subsequences}},
author = {Jerzy Kąkol},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17432},
eprint = {2609.17432},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17432},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Normal sequences without strongly normal subsequences
AU - Jerzy Kąkol
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17432
UR - https://arxiv.org/abs/2609.17432
AB - A sequence (y_n^*) in the unit sphere of E^* is strongly normal if the vectors x∈ E satisfying ∑_n|y_n^*(x)|<∞ form a dense subspace. For every set Γ of cardinality at least the continuum, we construct a normalized weakly null sequence in ℓ_1(Γ)^* with no strongly normal subsequence. This answers a question of Śliwa negatively in ZFC. Together with the classical selection argument below the bounding number b, the construction gives positive and negative bounds for this subsequence property on ℓ_1(κ). Its validity on ℓ_1(ω_1) is independent of ZFC. The intermediate range bleqslantκ<c remains undecided by these results. We also give a counterexample in a Banach sequence space that is not isomorphic to any ℓ_1(Γ).
ER -