Lotz-Peck-Porta and Rosenthal's theorems for spaces C_p(X)

2025/09/10 by Kąkol, Jerzy, Kurka, Ondřej, Śliwa, Wiesław
#46A03 #54C35 #54G12 #54H05 #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN)

paper · doi:10.48550/arxiv.2509.08634

Abstract

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 c₀ (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 c₀. 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 c₀(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.

Citations

• Grothendieck C(K)-spaces and the Josefson--Nissenzweig theorem
• When is a locally convex space Eberlein-Grothendieck?
• On linear continuous operators between distinguished spaces C_p(X)

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