## Abstract

C_{p}(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 E_{w}. It is unknown whether C_{p}(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 E_{w} is homeomorphic to the space C_{p}(X) for some infinite Tychonoff space X? We show that if such homeomorphism exists, then (a) X is a countable union of compact sets X_{n}, n∈ ω, where at least one component X_{n} is non-scattered; (b) the Banach space E necessarily contains an isomorphic copy of the Banach space ℓ_{1}.

Original language | American English |
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Article number | 150 |

Journal | Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas |

Volume | 116 |

Issue number | 4 |

DOIs | |

State | Published - 1 Oct 2022 |

## Keywords

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

## All Science Journal Classification (ASJC) codes

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

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