Distinguished Cp(X) spaces

J. C. Ferrando; J. Ka̧kol; A. Leiderman; S. A. Saxon

doi:10.1007/s13398-020-00967-4

Distinguished C_p(X) spaces

J. C. Ferrando, J. Ka̧kol, A. Leiderman, S. A. Saxon

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

15 Scopus citations

Abstract

We continue our initial study of C_p(X) spaces that are distinguished, equiv., are large subspaces of R^X, equiv., whose strong duals L_β(X) carry the strongest locally convex topology. Many are distinguished, many are not. All L_β(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_β(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 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.

Original language	English
Article number	27
Journal	Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas
Volume	115
Issue number	1
DOIs	https://doi.org/10.1007/s13398-020-00967-4
State	Published - 1 Jan 2021

Keywords

Bidual space
Distinguished space
Eberlein compact space
Fréchet space
Fundamental family of bounded sets
G-dense subspace
Point-finite family
strongly splittable space

ASJC Scopus subject areas

Analysis
Algebra and Number Theory
Geometry and Topology
Computational Mathematics
Applied Mathematics

Access to Document

10.1007/s13398-020-00967-4

Cite this

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title = "Distinguished Cp(X) spaces",

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

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author = "Ferrando, {J. C.} and J. K{\c a}kol and A. Leiderman and Saxon, {S. A.}",

note = "Funding Information: 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 GA{\v C}R Project 20-22230L and RVO: 67985840. Publisher Copyright: {\textcopyright} 2020, The Royal Academy of Sciences, Madrid.",

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AU - Ka̧kol, J.

AU - Leiderman, A.

AU - Saxon, S. A.

N1 - Funding Information: 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. Publisher Copyright: © 2020, The Royal Academy of Sciences, Madrid.

PY - 2021/1/1

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N2 - We continue our initial study of Cp(X) spaces that are distinguished, equiv., are large subspaces of RX, equiv., whose strong duals Lβ(X) carry the strongest locally convex topology. Many are distinguished, many are not. All Lβ(X) spaces are, as are all metrizable Cp(X) and Ck(X) spaces. To prove a space Cp(X) is not distinguished, we typically compare the character of Lβ(X) with |X|. A certain covering for X we call a scant cover is used to find distinguished Cp(X) spaces. Two of the main results are: (i) Cp(X) is distinguished if and only if its bidual E coincides with RX, and (ii) for a Corson compact space X, the space Cp(X) is distinguished if and only if X is scattered and Eberlein compact.

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

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