2021/09/13 by Jerzy Kcakol, Kcakol, Jerzy, Arkady Leiderman +3
Mathematics · #46B04 #46E10 #46E15 #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #Functional Equations Stability Results #General Topology (math.GN) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2109.06338
openalex publication_date 2021/09/13 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
Cp(X) denotes the space of continuous real-valued functions on a Tychonoff\nspace X endowed with the topology of pointwise convergence. A Banach space\nE equipped with the weak topology is denoted by Ew. It is unknown\nwhether Cp(K) and C(L)w can be homeomorphic for infinite compact spaces\nK and L citeKrupski-1, citeKrupski-2. In this paper we deal with a\nmore general question: what are the Banach spaces E which admit certain\ncontinuous surjective mappings T: Cp(X) \→ Ew for an infinite Tychonoff\nspace X?\n First, we prove that if T is linear and sequentially continuous, then the\nBanach space E must be finite-dimensional, thereby resolving an open problem\nposed in citeKakol-Leiderman. Second, we show that if there exists a\nhomeomorphism T: Cp(X) \→ Ew for some infinite Tychonoff space X and a\nBanach space E, then (a) X is a countable union of compact sets Xn, n \∈\n\ω, where at least one component Xn is non-scattered; (b) E\nnecessarily contains an isomorphic copy of the Banach space \ℓ1.\n