Distinguished Cp(X) spaces

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

doi:https://doi.org/10.1007/s13398-020-00967-4

Distinguished C_p(X) spaces

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

Department of Mathematics

Ben-Gurion University of the Negev

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

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	American 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

All Science Journal Classification (ASJC) codes

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

Access to Document

https://doi.org/10.1007/s13398-020-00967-4

Cite this

@article{dfaff9c8d894413b9d9ef86424f9287f,

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.",

keywords = "Bidual space, Distinguished space, Eberlein compact space, Fr{\'e}chet space, Fundamental family of bounded sets, G-dense subspace, Point-finite family, strongly splittable space",

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

note = "Publisher Copyright: {\textcopyright} 2020, The Royal Academy of Sciences, Madrid.",

year = "2021",

month = jan,

day = "1",

doi = "https://doi.org/10.1007/s13398-020-00967-4",

language = "American English",

volume = "115",

journal = "Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas",

issn = "1578-7303",

publisher = "Springer-Verlag Italia s.r.l.",

number = "1",

}

TY - JOUR

T1 - Distinguished Cp(X) spaces

AU - Ferrando, J. C.

AU - Ka̧kol, J.

AU - Leiderman, A.

AU - Saxon, S. A.

PY - 2021/1/1

Y1 - 2021/1/1

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.

KW - Bidual space

KW - Distinguished space

KW - Eberlein compact space

KW - Fréchet space

KW - Fundamental family of bounded sets

KW - G-dense subspace

KW - Point-finite family

KW - strongly splittable space

UR - http://www.scopus.com/inward/record.url?scp=85096666805&partnerID=8YFLogxK

U2 - https://doi.org/10.1007/s13398-020-00967-4

DO - https://doi.org/10.1007/s13398-020-00967-4

M3 - Article

SN - 1578-7303

VL - 115

JO - Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas

JF - Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas

IS - 1

M1 - 27

ER -