A note on Banach spaces E for which Ew is homeomorphic to Cp(X)

Jerzy Ka̧kol, Arkady Leiderman, Artur Michalak

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


Cp(X) denotes the space of continuous real-valued functions on a Tychonoff space X endowed with the topology of pointwise convergence. A Banach space E equipped with the weak topology is denoted by Ew. It is unknown whether Cp(K) and C(L) w can be homeomorphic for infinite compact spaces K and L (Krupski, Rev R Acad Cienc Exact Fis Nat Ser A Mat (RACSAM) 110:557–563, 2016; Krupski and Marciszewski, J Math Anal Appl 452:646–658, 2017). In this paper we deal with a more general question: does there exist a Banach space E such that Ew is homeomorphic to the space Cp(X) for some infinite Tychonoff space X? We show that if such homeomorphism exists, then (a) X is a countable union of compact sets Xn, n∈ ω, where at least one component Xn is non-scattered; (b) the Banach space E necessarily contains an isomorphic copy of the Banach space ℓ1.

Original languageEnglish
Article number150
JournalRevista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas
Issue number4
StatePublished - 1 Oct 2022


  • Banach space
  • C(X) space
  • Homeomorphism
  • Weak topology

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology
  • Computational Mathematics
  • Applied Mathematics


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