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
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Czech description
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Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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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
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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