2023/11/19 by Artur Bartoszewicz, Bartoszewicz, Artur, Szymon Głcab +1
Mathematics · #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA)
paper · pdf · doi:10.48550/arxiv.2311.11414
openalex publication_date 2023/11/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper is inspired by the paper of Leonetti, Russo and Somaglia [Dense lineability and spaceability in certain subsets of ℓ_∞. Bull. London Math. Soc., 55: 2283--2303 (2023)] and the lineability problems raised therein. It concerns the properties of ℓ_∞ subsets defined by cluster points of sequences. Using the fact that the set of cluster points of a sequence x depends only on its equivalence class in ℓ_∞/c0 and that the quotient space ℓ_∞/c0 is isometrically isomorphic to C(βℕ∖ℕ), we are able to translate lineability problems from ℓ_∞ to C(βℕ∖ℕ). We prove that for a compact space K with properties similar to those of βℕ∖ℕ, the sets of continuous functions f in C(K) with \vertrng(f)\vert=ω and those f with \vertrng(f)\vert=\mathfrak c contain, up to zero function, an isometric copy of c0(κ) for uncountable cardinal κ. Specializing those results to some closed subspaces K of βℕ∖ℕ we are able to generalize known results to their ideal versions.