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Witnessing the lack of the Grothendieck property in C(K)-spaces via convergent sequences

The result's identifiers

  • Result code in IS VaVaI

    <a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F20%3A00531560" target="_blank" >RIV/67985840:_____/20:00531560 - isvavai.cz</a>

  • Result on the web

    <a href="https://doi.org/10.1007/s13398-020-00914-3" target="_blank" >https://doi.org/10.1007/s13398-020-00914-3</a>

  • DOI - Digital Object Identifier

    <a href="http://dx.doi.org/10.1007/s13398-020-00914-3" target="_blank" >10.1007/s13398-020-00914-3</a>

Alternative languages

  • Result language

    angličtina

  • Original language name

    Witnessing the lack of the Grothendieck property in C(K)-spaces via convergent sequences

  • Original language description

    Let K be a compact Hausdorff space and let C(K) be the space of all scalar-valued, continuous functions on K. We show that C(K) is an ℓ1(K) -Grothendieck space but not a Grothendieck space exactly when the spaces Cp(K) and Cp(K⊕ N#) are not linearly isomorphic, where N# is the one-point compactificiation of the discrete space of natural numbers. (That is, if C(K) contains a complemented copy of c, then C(K) fails to be ℓ1(K) -Grothendieck if and only if the topologies of pointwise convergence in Cp(K) and Cp(K⊕ N#) are linearly isomorphic.) Moreover, for infinite compact spaces K and L, there exists a compact space G that has a non-trivial convergent sequence and such that Cp(K× L) and Cp(G) are linearly isomorphic. This extends a remarkable theorem of Cembranos and Freniche. Some examples illustrating the above results are provided.

  • Czech name

  • Czech description

Classification

  • Type

    J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database

  • CEP classification

  • OECD FORD branch

    10101 - Pure mathematics

Result continuities

  • Project

    <a href="/en/project/GF20-22230L" target="_blank" >GF20-22230L: Banach spaces of continuous and Lipschitz functions</a><br>

  • Continuities

    I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Others

  • Publication year

    2020

  • Confidentiality

    S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Data specific for result type

  • Name of the periodical

    Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales

  • ISSN

    1578-7303

  • e-ISSN

  • Volume of the periodical

    114

  • Issue of the periodical within the volume

    4

  • Country of publishing house

    ES - SPAIN

  • Number of pages

    7

  • Pages from-to

    179

  • UT code for WoS article

    000555465400001

  • EID of the result in the Scopus database

    2-s2.0-85088986527