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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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If not distinguished, is $C_{p}\left ( X\right )$ even close?
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by J. C. Ferrando and Stephen A. Saxon PDF
Proc. Amer. Math. Soc. 149 (2021), 2583-2596 Request permission


$C_{p}\left ( X\right )$ 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
  • Email:
  • Stephen A. Saxon
  • Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611
  • MR Author ID: 155275
  • Email:
  • 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
  • © Copyright 2021 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 149 (2021), 2583-2596
  • MSC (2020): Primary 46A08, 54C35
  • DOI:
  • MathSciNet review: 4246809