{
  "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 $\\ell_{\\infty}^{*}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12787",
  "author_total": 1,
  "authors": [
    {
      "name": "Wiesław Śliwa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3vgrqvcvnhd0n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12787",
  "fulltext_url": null,
  "id": "3vgrqvcvnhd0n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12787",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7221516831",
  "pdf_url": "https://arxiv.org/pdf/2610.12787",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3vgrqvcvnhd0n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.12787",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12787"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Reflexivity of Grothendieck Spaces and Strongly Normal Sequences",
  "title_text": "Reflexivity of Grothendieck Spaces and Strongly Normal Sequences",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/3vgrqvcvnhd0n",
  "venue": "arXiv",
  "versions": []
}