Linear continuous surjections of C_p-spaces over compacta

2016/05/17 by Kazuhiro Kawamura, Kawamura, Kazuhiro, Arkady Leiderman +1
Mathematics · #46E10 #54C35 #54F45 #Advanced Banach Space Theory #Advanced Topology and Set Theory #Approximation Theory and Sequence Spaces #FOS: Mathematics #General Topology (math.GN)

paper · pdf · doi:10.48550/arxiv.1605.05276

openalex publication_date 2016/05/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X and Y be compact Hausdorff spaces and suppose that there exists a linear continuous surjection T:C_p(X) → C_p(Y), where C_p(X) denotes the space of all real-valued continuous functions on X endowed with the pointwise convergence topology. We prove that dim X=0 implies dim Y = 0. This generalizes a previous theorem \cite[Theorem 3.4]LLP for compact metrizable spaces. Also we point out that the function space C_p(P) over the pseudo-arc P admits no densely defined linear continuous operator C_p(P) → C_p([0,1]) with a dense image.

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