Note on the Banach Problem 1 of condensations of Banach spaces onto compacta

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Note on the Banach Problem 1 of condensations of Banach spaces onto compacta. / Osipov, A. V.
In: Filomat, Vol. 37, No. 7, 2023, p. 2183-2186.

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

BibTeX

@article{b9d39950f49e43ec971061e118e2d9a8,

title = "Note on the Banach Problem 1 of condensations of Banach spaces onto compacta",

abstract = "It is consistent with any possible value of the continuum c that every infinite-dimensional Banach space of density ≤ c condenses onto the Hilbert cube. Let µ < c be a cardinal of uncountable cofinality. It is consistent that the continuum be arbitrary large, no Banach space X of density γ, µ < γ < c, condenses onto a compact metric space, but any Banach space of density µ admits a condensation onto a compact metric space. In particular, for µ = ω1, it is consistent that c is arbitrarily large, no Banach space of density γ, ω1 < γ < c, condenses onto a compact metric space. These results imply a complete answer to the Problem 1 in the Scottish Book for Banach spaces: When does a Banach space X admit a bijective continuous mapping onto a compact metric space?. ",

author = "Osipov, {A. V.}",

year = "2023",

doi = "10.2298/FIL2307183O",

language = "English",

volume = "37",

pages = "2183--2186",

journal = "Filomat",

issn = "0354-5180",

publisher = "Universitet of Nis",

number = "7",

}

RIS

TY - JOUR

T1 - Note on the Banach Problem 1 of condensations of Banach spaces onto compacta

AU - Osipov, A. V.

PY - 2023

Y1 - 2023

N2 - It is consistent with any possible value of the continuum c that every infinite-dimensional Banach space of density ≤ c condenses onto the Hilbert cube. Let µ < c be a cardinal of uncountable cofinality. It is consistent that the continuum be arbitrary large, no Banach space X of density γ, µ < γ < c, condenses onto a compact metric space, but any Banach space of density µ admits a condensation onto a compact metric space. In particular, for µ = ω1, it is consistent that c is arbitrarily large, no Banach space of density γ, ω1 < γ < c, condenses onto a compact metric space. These results imply a complete answer to the Problem 1 in the Scottish Book for Banach spaces: When does a Banach space X admit a bijective continuous mapping onto a compact metric space?.

AB - It is consistent with any possible value of the continuum c that every infinite-dimensional Banach space of density ≤ c condenses onto the Hilbert cube. Let µ < c be a cardinal of uncountable cofinality. It is consistent that the continuum be arbitrary large, no Banach space X of density γ, µ < γ < c, condenses onto a compact metric space, but any Banach space of density µ admits a condensation onto a compact metric space. In particular, for µ = ω1, it is consistent that c is arbitrarily large, no Banach space of density γ, ω1 < γ < c, condenses onto a compact metric space. These results imply a complete answer to the Problem 1 in the Scottish Book for Banach spaces: When does a Banach space X admit a bijective continuous mapping onto a compact metric space?.

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U2 - 10.2298/FIL2307183O

DO - 10.2298/FIL2307183O

M3 - Article

VL - 37

SP - 2183

EP - 2186

JO - Filomat

JF - Filomat

SN - 0354-5180

IS - 7

ER -

ID: 35499357