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

0559489 - MÚ 2023 RIV ES eng J - Journal Article
Kąkol, Jerzy - Leiderman, A. G. - Michalak, A.
A note on Banach spaces E for which Ew is homeomorphic to Cp(X).
Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Roč. 116, č. 4 (2022), č. článku 150. ISSN 1578-7303. E-ISSN 1579-1505
R&D Projects: GA ČR(CZ) GF20-22230L
Institutional support: RVO:67985840
Keywords : Banach space * homeomorphism * weak topology
OECD category: Pure mathematics
Impact factor: 2.9, year: 2022
Method of publishing: Limited access
https://doi.org/10.1007/s13398-022-01292-8

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.
Permanent Link: https://hdl.handle.net/11104/0332769