Lotz-Peck-Porta and Rosenthal's theorems for spaces $C_p(X)$
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arXiv
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| Format: | Preprint |
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2025
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| author | Kąkol, Jerzy Kurka, Ondřej Śliwa, Wiesław |
| author_facet | Kąkol, Jerzy Kurka, Ondřej Śliwa, Wiesław |
| contents | For a Tychonoff space $X$ by $C_p(X)$ we denote the space $C(X)$ of continuous real valued functions on $X$ endowed with the pointwise topology. We prove that an infinite compact space $X$ is scattered if and only if every closed infinite-dimensional subspace in $C_p(X)$ contains a copy of $c_0$ (with the pointwise topology) which is complemented in the whole space $C_p(X)$. This provides a $C_p$-version of the theorem of Lotz, Peck and Porta for Banach spaces $C(X)$ and $c_0$. Applications will be provided. We prove also a $C_p$-version of Rosenthal's theorem by showing that for an infinite compact $X$ the space $C_p(X)$ contains a closed copy of $c_{0}(Γ)$ (with the pointwise topology) for some uncountable set $Γ$ if and only if $X$ admits an uncountable family of pairwise disjoint open subsets of $X$. Illustrating examples, additional supplementing $C_p$-theorems and comments are included. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_08634 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lotz-Peck-Porta and Rosenthal's theorems for spaces $C_p(X)$ Kąkol, Jerzy Kurka, Ondřej Śliwa, Wiesław Functional Analysis General Topology 54C35, 54G12, 54H05, 46A03 For a Tychonoff space $X$ by $C_p(X)$ we denote the space $C(X)$ of continuous real valued functions on $X$ endowed with the pointwise topology. We prove that an infinite compact space $X$ is scattered if and only if every closed infinite-dimensional subspace in $C_p(X)$ contains a copy of $c_0$ (with the pointwise topology) which is complemented in the whole space $C_p(X)$. This provides a $C_p$-version of the theorem of Lotz, Peck and Porta for Banach spaces $C(X)$ and $c_0$. Applications will be provided. We prove also a $C_p$-version of Rosenthal's theorem by showing that for an infinite compact $X$ the space $C_p(X)$ contains a closed copy of $c_{0}(Γ)$ (with the pointwise topology) for some uncountable set $Γ$ if and only if $X$ admits an uncountable family of pairwise disjoint open subsets of $X$. Illustrating examples, additional supplementing $C_p$-theorems and comments are included. |
| title | Lotz-Peck-Porta and Rosenthal's theorems for spaces $C_p(X)$ |
| topic | Functional Analysis General Topology 54C35, 54G12, 54H05, 46A03 |
| url | https://arxiv.org/abs/2509.08634 |