{
  "abstract": "We study conditions that imply finite-dimensionality of metrizable or Frechet-Urysohn linear subspaces of topological vector spaces (TVS). We consider the classes $\\mathscr{D}_M$ and $\\mathscr{D}_{FU}$ of TVS: a space $L$ belongs to $\\mathscr{D}_M$ (respectively, $\\mathscr{D}_{FU}$) if every metrizable (respectively, Frechet-Urysohn) linear subspace of $L$ is finite-dimensional. Although the classes of metrizable spaces and of Frechet-Urysohn spaces differ substantially, the classes $\\mathscr{D}_M$ and $\\mathscr{D}_{FU}$ are much closer to each other: in the Hrusak-Ramos-Garcia model, in which Malykhin's problem is solved, $\\mathscr{D}_M=\\mathscr{D}_{FU}$, whereas under MA+not CH, as shown in the paper, $\\mathscr{D}_M\\neq\\mathscr{D}_{FU}$. For a locally convex space (LCS) $L$ we denote by $L_w$ the space $L$ with its weak topology and by $L'_{w^*}$ the dual space with the weak$^*$ topology. This paper investigates which LCS $L$ have the property that $L_w$ and $L'_{w^*}$ belong to $\\mathscr{D}_M$ or $\\mathscr{D}_{FU}$. Although the classes are close, different methods are used to study them: topological-algebraic ones for $\\mathscr{D}_M$ (a criterion is obtained) and methods of measure theory and probability for $\\mathscr{D}_{FU}$ (a sufficient condition is obtained). We prove that if an LCS $L$ admits a continuous norm, then $L_w\\in\\mathscr{D}_{FU}$; in particular, if $L$ admits a continuous norm and $L_w$ is Frechet-Urysohn, then $L$ is finite-dimensional. If the topology of an LCS $L$ is coarser than the topology induced by some complete norm on $L$ (for example, if $L$ is Banach), then $L'_{w^*}\\in\\mathscr{D}_{FU}$; completeness of the norm is essential here. The methods also apply to spaces of real-analytic functions and to the space of polynomials with the topology of pointwise convergence: these also belong to $\\mathscr{D}_{FU}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.30800",
  "author_total": 2,
  "authors": [
    {
      "name": "Vladislav V. Maksimov"
    },
    {
      "name": "Evgenii A. Reznichenko"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b0ya2yjnjqhs5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.30800",
  "fulltext_url": null,
  "id": "b0ya2yjnjqhs5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.30800",
  "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"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7223373396",
  "pdf_url": "https://arxiv.org/pdf/2610.30800",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b0ya2yjnjqhs5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.30800",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.30800"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    }
  ],
  "title": "When are metrizable linear subspaces of topological vector spaces finite-dimensional?",
  "title_text": "When are metrizable linear subspaces of topological vector spaces finite-dimensional?",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/b0ya2yjnjqhs5",
  "venue": "arXiv",
  "versions": []
}