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| Main Authors: | , |
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| Format: | Preprint |
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2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2311.11414 |
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| _version_ | 1866914644779597824 |
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| author | Bartoszewicz, Artur Głcab, Szymon |
| author_facet | Bartoszewicz, Artur Głcab, Szymon |
| contents | This paper is inspired by the paper of Leonetti, Russo and Somaglia [\textit{Dense lineability and spaceability in certain subsets of $\ell_\infty$.} Bull. London Math. Soc., 55: 2283--2303 (2023)] and the lineability problems raised therein. It concerns the properties of $\ell_\infty$ 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 $\ell_\infty/c_0$ and that the quotient space $\ell_\infty/c_0$ is isometrically isomorphic to $C(β\mathbb{N}\setminus\mathbb{N})$, we are able to translate lineability problems from $\ell_\infty$ to $C(β\mathbb{N}\setminus\mathbb{N})$. We prove that for a compact space $K$ with properties similar to those of $β\mathbb{N}\setminus\mathbb{N}$, the sets of continuous functions $f$ in $C(K)$ with $\vert\operatorname{rng}(f)\vert=ω$ and those $f$ with $\vert\operatorname{rng}(f)\vert=\mathfrak c$ contain, up to zero function, an isometric copy of $c_0(κ)$ for uncountable cardinal $κ$. Specializing those results to some closed subspaces $K$ of $β\mathbb{N}\setminus\mathbb{N}$ we are able to generalize known results to their ideal versions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_11414 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Lineability of functions in $C(K)$ with specified range Bartoszewicz, Artur Głcab, Szymon Functional Analysis This paper is inspired by the paper of Leonetti, Russo and Somaglia [\textit{Dense lineability and spaceability in certain subsets of $\ell_\infty$.} Bull. London Math. Soc., 55: 2283--2303 (2023)] and the lineability problems raised therein. It concerns the properties of $\ell_\infty$ 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 $\ell_\infty/c_0$ and that the quotient space $\ell_\infty/c_0$ is isometrically isomorphic to $C(β\mathbb{N}\setminus\mathbb{N})$, we are able to translate lineability problems from $\ell_\infty$ to $C(β\mathbb{N}\setminus\mathbb{N})$. We prove that for a compact space $K$ with properties similar to those of $β\mathbb{N}\setminus\mathbb{N}$, the sets of continuous functions $f$ in $C(K)$ with $\vert\operatorname{rng}(f)\vert=ω$ and those $f$ with $\vert\operatorname{rng}(f)\vert=\mathfrak c$ contain, up to zero function, an isometric copy of $c_0(κ)$ for uncountable cardinal $κ$. Specializing those results to some closed subspaces $K$ of $β\mathbb{N}\setminus\mathbb{N}$ we are able to generalize known results to their ideal versions. |
| title | Lineability of functions in $C(K)$ with specified range |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2311.11414 |