Linear continuous surjections of $C_{p}$-spaces over compacta
Kazuhiro Kawamura, Arkady Leiderman

TL;DR
This paper investigates the structure of linear continuous surjections between function spaces over compact spaces, showing that zero-dimensionality is preserved and exploring operator limitations over the pseudo-arc.
Contribution
It generalizes previous results by proving zero-dimensionality preservation under surjections for broader classes of compact spaces and examines operator restrictions over the pseudo-arc.
Findings
Zero-dimensionality is preserved under surjective linear continuous maps.
No densely defined linear continuous operator with dense image exists from the pseudo-arc space to the unit interval.
The result extends known theorems from metrizable to general compact spaces.
Abstract
Let and be compact Hausdorff spaces and suppose that there exists a linear continuous surjection , where denotes the space of all real-valued continuous functions on endowed with the pointwise convergence topology. We prove that implies . This generalizes a previous theorem \cite[Theorem 3.4]{LLP} for compact metrizable spaces. Also we point out that the function space over the pseudo-arc admits no densely defined linear continuous operator with a dense image.
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Taxonomy
TopicsAdvanced Topology and Set Theory · Advanced Banach Space Theory · Approximation Theory and Sequence Spaces
