The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F16%3A00460235" target="_blank" >RIV/67985840:_____/16:00460235 - isvavai.cz</a>
Result on the web
<a href="http://dx.doi.org/10.4064/sm8289-4-2016" target="_blank" >http://dx.doi.org/10.4064/sm8289-4-2016</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.4064/sm8289-4-2016" target="_blank" >10.4064/sm8289-4-2016</a>
Alternative languages
Result language
angličtina
Original language name
The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces
Original language description
Following Banakh and Gabriyelyan (2016) we say that a Tychonoff space X is an Ascoli space if every compact subset K of C-k (X) is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every k(R)-space, hence any k-space, is Ascoli. Let X be a metrizable space. We prove that the space C-k (X) is Ascoli iff C-k (X) is a k(R)-space iff X is locally compact. Moreover, C-k (X) endowed with the weak topology is Ascoli iff X is countable and discrete. Using some basic concepts from probability theory and measure-theoretic properties of l(1), we show that the following assertions are equivalent for a Banach space E : (i) E does not contain an isomorphic copy of l(1), (ii) every real-valued sequentially continuous map on the unit ball B-w with the weak topology is continuous, (iii) B-w is a k(R)-space, (iv) B-w is an Ascoli space.
Czech name
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Czech description
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Classification
Type
J<sub>x</sub> - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
BA - General mathematics
OECD FORD branch
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Result continuities
Project
<a href="/en/project/GF16-34860L" target="_blank" >GF16-34860L: Logic and Topology in Banach spaces</a><br>
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2016
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Studia mathematica
ISSN
0039-3223
e-ISSN
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Volume of the periodical
233
Issue of the periodical within the volume
2
Country of publishing house
PL - POLAND
Number of pages
21
Pages from-to
119-139
UT code for WoS article
000376610400002
EID of the result in the Scopus database
2-s2.0-84973517393