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Distinguished \( C_{p}(X) \) spaces

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

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Abstract

We continue our initial study of \(C_{p}(X) \) spaces that are distinguished, equiv., are large subspaces of \(\mathbb {R}^{X}\), equiv., whose strong duals \(L_{\beta }( X) \) carry the strongest locally convex topology. Many are distinguished, many are not. All \(L_{\beta }(X) \) spaces are, as are all metrizable \(C_{p}(X) \) and \( C_{k} ( X) \) spaces. To prove a space \(C_{p}(X) \) is not distinguished, we typically compare the character of \(L_{\beta }(X) \) with |X|. A certain covering for X we call a scant cover is used to find distinguished \(C_{p} ( X) \) spaces. Two of the main results are: (i) \(C_{p}(X) \) is distinguished if and only if its bidual E coincides with \(\mathbb {R}^{X}\), and (ii) for a Corson compact space X, the space \( C_{p}(X) \) is distinguished if and only if X is scattered and Eberlein compact.

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Change history

  • 13 January 2021

    Unfortunately, the reference 19 is updated incorrectly in the original article and the reference 19 should be.

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Acknowledgements

We wish to thank Vladimir Tkachuk for providing us Example 32 and Witold Marciszewski for references [1, 6] as well as helpful discussion on scattered Eberlein compact spaces.

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Authors and Affiliations

  1. Centro de Investigación Operativa, Universidad Miguel Hernández, 03202, Elche, Spain

    J. C. Ferrando

  2. Faculty of Mathematics and Informatics, A. Mickiewicz University, 61-614, Poznan, Poland

    J. Ka̧kol

  3. Institute of Mathematics Czech Academy of Sciences, Prague, Czechia

    J. Ka̧kol

  4. Department of Mathematics, Ben-Gurion University of Negev, Beer Sheva, Israel

    A. Leiderman

  5. Department of Mathematics, University of Florida, Gainesville, FL, 32611, USA

    S. A. Saxon

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  1. J. C. Ferrando
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  2. J. Ka̧kol
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Correspondence to J. C. Ferrando.

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The first named author is supported by Grant PGC2018-094431-B-I00 of the Ministry of Science, Innovation and Universities of Spain. The research for the second named author is supported by the GAČR Project 20-22230L and RVO: 67985840.

The original version of this article was revised: Reference 19 was corrected.

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Ferrando, J.C., Ka̧kol, J., Leiderman, A. et al. Distinguished \( C_{p}(X) \) spaces. RACSAM 115, 27 (2021). https://doi.org/10.1007/s13398-020-00967-4

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