For a Tychonoff space X by <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$C_p(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$C_p(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> contains a copy of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$c_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>c</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> (with the pointwise topology) which is complemented in the whole space <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$C_p(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. This provides a <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$C_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>-version of the theorem of Lotz, Peck and Porta for Banach spaces C(X) and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$c_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>c</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>. Applications will be provided. We prove also a <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$C_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>-version of Rosenthal’s theorem by showing that for an infinite compact X the space <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$C_p(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> contains a closed copy of <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$c_{0}(\Gamma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>c</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> (with the pointwise topology) for some uncountable set <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> if and only if X admits an uncountable family of pairwise disjoint open subsets of X. Illustrating examples, additional supplementing <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$C_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>-theorems and comments are included.

Lotz–Peck–Porta and Rosenthal’s theorems for spaces $C_p(X)$

期刊论文
发表日期：

2026年2月4日

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Jerzy Ka̧kol,
Ondřej Kurka,
Wiesław Śliwa

摘要

For a Tychonoff space X by $C_p(X)$ $C_{p} (X)$ 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 $C_p(X)$ $C_{p} (X)$ contains a copy of $c_0$ $c_{0}$ (with the pointwise topology) which is complemented in the whole space $C_p(X)$ $C_{p} (X)$ . This provides a $C_p$ $C_{p}$ -version of the theorem of Lotz, Peck and Porta for Banach spaces C(X) and $c_0$ $c_{0}$ . Applications will be provided. We prove also a $C_p$ $C_{p}$ -version of Rosenthal’s theorem by showing that for an infinite compact X the space $C_p(X)$ $C_{p} (X)$ contains a closed copy of $c_{0}(\Gamma )$ $c_{0} (Γ)$ (with the pointwise topology) for some uncountable set $\Gamma $ $Γ$ if and only if X admits an uncountable family of pairwise disjoint open subsets of X. Illustrating examples, additional supplementing $C_p$ $C_{p}$ -theorems and comments are included.

Lotz–Peck–Porta and Rosenthal’s theorems for spaces \(C_p(X)\)

摘要