{
  "abstract": "In 1940, Levin\\footnotemark[1] and Milman proved that a closed linear subspace of $C[0,1]$ whose elements all have bounded Jordan variation must be finite-dimensional. We prove its analogue for variation in the sense of Young and, more generally, for every finite-valued nondecreasing gauge $φ:[0,\\infty)\\to[0,\\infty)$ with $φ(0)=0<φ(t)$ for $t>0$. If $E$ is a closed linear subspace of $C[0,1]$ and every $f\\in E$ satisfies $\\Var_φ(λ_f f)<\\infty$ at some scale $λ_f>0$, then $E$ is finite-dimensional. No continuity, convexity, or doubling condition is needed. The proof combines a lower-semicontinuous regularization that preserves the scaled class, Baire uniformization, Helly selection, and a quantitative nested-peaks construction for nonhomogeneous gauges. Consequently, for every infinite-dimensional closed subspace $F\\subset C[0,1]$, the set $F\\cap\\cV_φ([0,1])$ is a meagre $F_σ$ subset of $F$. For every Young function, both the scaled and raw finite-variation families are maximal dense-lineable but not spaceable. The scaled family is itself a dense vector subspace of Hamel dimension $\\mathfrak c$, whereas without local doubling the raw family need not itself be linear.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.07132",
  "author_total": 4,
  "authors": [
    {
      "name": "N. G. Albuquerque"
    },
    {
      "name": "D. Bugajewska"
    },
    {
      "name": "E. Demétrio Jr."
    },
    {
      "name": "J. B. Seoane-Sepúlveda"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6bhyzwvm96dpx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.07132",
  "fulltext_url": null,
  "id": "6bhyzwvm96dpx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.07132",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7212073197",
  "pdf_url": "https://arxiv.org/pdf/2609.07132",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6bhyzwvm96dpx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.07132",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.07132"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A Levin-Milman theorem for Young variation",
  "title_text": "A Levin-Milman theorem for Young variation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/6bhyzwvm96dpx",
  "venue": "arXiv",
  "versions": []
}