Skip to content
DraconiticResearch
Search Sources API
181,713 studies31 sourcesUpdated 21 min ago
PreprintarXiv licencearXiv

On condensations of Cp(X) spaces onto Banach spaces with the weak topology

  • Mikołaj Krupski
  • Witold Marciszewski
Open at sourcePDFDOICite

Abstract

We show that for a broad class of spaces X, the space Cp(X) of continuous real-valued functions on X equipped with the pointwise topology maps bijectively onto any separable infinite-dimensional Banach space with the weak topology. We prove that the inverse φ−1 of any continuous bijection φ: Cp(X)→ (E,w), where X is an arbitrary infinite Tychonoff space and (E,w) is an arbitrary normed space with the weak topology, is everywhere discontinuous. This generalizes a recent result of Kania--Kąkol and solves a problem raised by Kąkol, Leiderman and Michalak by providing a complete classification of function spaces Cp(X) that are homeomorphic to Banach spaces with the weak topology.

Related

  1. Shadowing and inverse shadowing in spaces retractable onto graphs
    Jonathan Meddaugh, Elyssa Stephens
    arXivPreprintCC BY 4.0Full textDynamical systemsarXiv
  2. Anisotropic error estimates for the L2-projection onto discontinuous finite element spaces
    Joaquín Valenzuela Díaz, Yuan Wang, Sander Rhebergen
    arXivPreprintCC BY 4.0Full textNumerical analysis & scientific computingarXiv
  3. Uniform Embeddings of Banach Space Spheres into Hilbert Spaces and Property (H)
    Qingjin Cheng, Yue Wang, Yipeng Zhang
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  4. Pointwise function spaces over compacta are not weak Banach spaces
    Tomasz Kania, Jerzy Kąkol
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsGeometry & topologyarXiv
  5. Improved Bounds For The Lipschitz Retraction Constant In Classical Banach Spaces
    Shriram Mahesh
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv

Cite

BibTeXRIS
BibTeX
@misc{krupski2026condensations,
  title = {{On condensations of $C_p(X)$ spaces onto Banach spaces with the weak topology}},
  author = {Mikołaj Krupski and Witold Marciszewski},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.24165},
  eprint = {2610.24165},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.24165},
}
RIS
TY  - UNPB
TI  - On condensations of C_p(X) spaces onto Banach spaces with the weak topology
AU  - Mikołaj Krupski
AU  - Witold Marciszewski
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.24165
UR  - https://arxiv.org/abs/2610.24165
AB  - We show that for a broad class of spaces X, the space C_p(X) of continuous real-valued functions on X equipped with the pointwise topology maps bijectively onto any separable infinite-dimensional Banach space with the weak topology. We prove that the inverse φ^−1 of any continuous bijection φ: C_p(X)→ (E,w), where X is an arbitrary infinite Tychonoff space and (E,w) is an arbitrary normed space with the weak topology, is everywhere discontinuous. This generalizes a recent result of Kania--Kąkol and solves a problem raised by Kąkol, Leiderman and Michalak by providing a complete classification of function spaces C_p(X) that are homeomorphic to Banach spaces with the weak topology.
ER  -