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PreprintCC BY 4.0arXiv

When are metrizable linear subspaces of topological vector spaces finite-dimensional?

  • Vladislav V. Maksimov
  • Evgenii A. Reznichenko
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Abstract

We study conditions that imply finite-dimensionality of metrizable or Frechet-Urysohn linear subspaces of topological vector spaces (TVS). We consider the classes π’ŸM and π’ŸFU of TVS: a space L belongs to π’ŸM (respectively, π’ŸFU) 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 π’ŸM and π’ŸFU are much closer to each other: in the Hrusak-Ramos-Garcia model, in which Malykhin's problem is solved, π’ŸM=π’ŸFU, whereas under MA+not CH, as shown in the paper, π’ŸMβ‰ π’ŸFU. For a locally convex space (LCS) L we denote by Lw 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 Lw and L'w* belong to π’ŸM or π’ŸFU. Although the classes are close, different methods are used to study them: topological-algebraic ones for π’ŸM (a criterion is obtained) and methods of measure theory and probability for π’ŸFU (a sufficient condition is obtained). We prove that if an LCS L admits a continuous norm, then Lwβˆˆπ’ŸFU; in particular, if L admits a continuous norm and Lw 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*βˆˆπ’ŸFU; 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 π’ŸFU.

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Cite

BibTeXRIS
BibTeX
@misc{maksimov2026when,
  title = {{When are metrizable linear subspaces of topological vector spaces finite-dimensional?}},
  author = {Vladislav V. Maksimov and Evgenii A. Reznichenko},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.30800},
  eprint = {2610.30800},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.30800},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - When are metrizable linear subspaces of topological vector spaces finite-dimensional?
AU  - Vladislav V. Maksimov
AU  - Evgenii A. Reznichenko
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.30800
UR  - https://arxiv.org/abs/2610.30800
AB  - We study conditions that imply finite-dimensionality of metrizable or Frechet-Urysohn linear subspaces of topological vector spaces (TVS). We consider the classes π’Ÿ_M and π’Ÿ_FU of TVS: a space L belongs to π’Ÿ_M (respectively, π’Ÿ_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 π’Ÿ_M and π’Ÿ_FU are much closer to each other: in the Hrusak-Ramos-Garcia model, in which Malykhin's problem is solved, π’Ÿ_M=π’Ÿ_FU, whereas under MA+not CH, as shown in the paper, π’Ÿ_Mβ‰ π’Ÿ_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 π’Ÿ_M or π’Ÿ_FU. Although the classes are close, different methods are used to study them: topological-algebraic ones for π’Ÿ_M (a criterion is obtained) and methods of measure theory and probability for π’Ÿ_FU (a sufficient condition is obtained). We prove that if an LCS L admits a continuous norm, then L_wβˆˆπ’Ÿ_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^*βˆˆπ’Ÿ_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 π’Ÿ_FU.
ER  -