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If not distinguished, is $C_{p}\left ( X\right )$ even close?

Authors: J. C. Ferrando and Stephen A. Saxon
Journal: Proc. Amer. Math. Soc. 149 (2021), 2583-2596
MSC (2020): Primary 46A08, 54C35
Published electronically: March 25, 2021
MathSciNet review: 4246809
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Abstract: $C_{p}\left ( X\right )$ is distinguished $\Leftrightarrow$ the strong dual $L_{\beta }\left ( X\right )$ is barrelled $\Leftrightarrow$ the strong bidual $M\left ( X\right ) =\mathbb {R}^{X}$. So one may judge how nearly distinguished $C_{p}\left ( X\right )$ is by how nearly barrelled $L_{\beta }\left ( X\right )$ is, and also by how near the dense subspace $M\left ( X\right )$ is to the Baire space $\mathbb {R}^{X}$. Being Baire-like, $M\left ( X\right )$ is always fairly close to $\mathbb {R}^{X}$ in that sense. But if $C_{p}\left ( X\right )$ is not distinguished, we show the codimension of $M\left ( X\right )$ is uncountable, i.e., $M\left ( X\right )$ is algebraically far from $\mathbb {R}^{X}$, and moreover, $L_{\beta }\left ( X\right )$ is very far from barrelled, not even primitive. Thus we profile weak barrelledness for $L_{\beta }\left ( X\right )$ and $M\left ( X\right )$ spaces. At the same time, we characterize those Tychonoff spaces $X$ for which $C_{p}\left ( X\right )$ is distinguished, solving the original problem from our series of papers.

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Additional Information

J. C. Ferrando
Affiliation: Centro de Investigación Operativa, Universidad Miguel Hernández, 03202 Elche, Spain
MR Author ID: 256880

Stephen A. Saxon
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611
MR Author ID: 155275

Keywords: Distinguished, $\aleph _{0}$-barrelled, primitive, weak barrelledness, bidual, $\Sigma \left ( X\right )$
Received by editor(s): April 24, 2020
Received by editor(s) in revised form: June 19, 2020, and December 4, 2020
Published electronically: March 25, 2021
Additional Notes: The first-named author was supported by Grant PGC2018-094431-B-I00 of the Ministry of Science, Innovation and Universities of Spain
Communicated by: Stephen Dilworth
Article copyright: © Copyright 2021 American Mathematical Society