{
  "abstract": "A sequence $(y_n^*)$ in the unit sphere of $E^*$ is strongly normal if the vectors $x\\in E$ satisfying $\\sum_n|y_n^*(x)|<\\infty$ form a dense subspace. For every set $Γ$ of cardinality at least the continuum, we construct a normalized weakly null sequence in $\\ell_1(Γ)^*$ with no strongly normal subsequence. This answers a question of Śliwa negatively in ZFC. Together with the classical selection argument below the bounding number $\\mathfrak b$, the construction gives positive and negative bounds for this subsequence property on $\\ell_1(κ)$. Its validity on $\\ell_1(ω_1)$ is independent of ZFC. The intermediate range $\\mathfrak b\\leqslantκ<\\mathfrak c$ remains undecided by these results. We also give a counterexample in a Banach sequence space that is not isomorphic to any $\\ell_1(Γ)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17432",
  "author_total": 1,
  "authors": [
    {
      "name": "Jerzy Kąkol"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0pd23ch18hh9k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17432",
  "fulltext_url": null,
  "id": "0pd23ch18hh9k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17432",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7213405890",
  "pdf_url": "https://arxiv.org/pdf/2609.17432",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0pd23ch18hh9k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.17432",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17432"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Normal sequences without strongly normal subsequences",
  "title_text": "Normal sequences without strongly normal subsequences",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/0pd23ch18hh9k",
  "venue": "arXiv",
  "versions": []
}