{
  "abstract": "In his 1994 doctoral thesis, Bossard introduced the notion of the non-universal tree $T_{NU}(X)$ associated with each separable Banach space $X$ which does not contain an isomorphic copy of $C(2^\\mathbb{N})$. Together with the order operation defined on well-founded trees, we obtain a method of classifying the complexity of separable Banach spaces by the degree of isomorphism of finite-dimensional subspaces of $C(2^\\mathbb{N})$. Despite further refinement of this concept in later years, we are unaware of any specific instances of direct exhibitions of the order of the non-universal tree for a concrete space and basis. We show that if $H$ is any separable Hilbert space, then $o(T_{NU}(H)) = ω + 1$ when taken with respect to the Haar basis for $C(2^\\mathbb{N})$, demonstrating that the class of Hilbert spaces is the least complex class with respect to this measurement when considering this basis.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01885",
  "author_total": 1,
  "authors": [
    {
      "name": "Sam Whitmire"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3man9yw8zg5ym.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01885",
  "fulltext_url": null,
  "id": "3man9yw8zg5ym",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01885",
  "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"
    ],
    [
      "arxiv",
      "math.GN"
    ],
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
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  "openalex": "W7207773522",
  "pdf_url": "https://arxiv.org/pdf/2609.01885",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3man9yw8zg5ym.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.01885",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01885"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "The Order of the Non-universal Tree of a Hilbert Space with Respect to the Haar Basis",
  "title_text": "The Order of the Non-universal Tree of a Hilbert Space with Respect to the Haar Basis",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/3man9yw8zg5ym",
  "venue": "arXiv",
  "versions": []
}