The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces

Popis výsledku anglicky

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.

Druh

J_x - Nezařazeno - Článek v odborném periodiku (Jimp, Jsc a Jost)

CEP obor

BA - Obecná matematika

OECD FORD obor

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Projekt

<a href="/cs/project/GF16-34860L" target="_blank" >GF16-34860L: Logika a topologie v Banachových prostorech</a><br>

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2016

Kód důvěrnosti údajů

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

Název periodika

Studia mathematica

ISSN

0039-3223

e-ISSN

—

Svazek periodika

233

Číslo periodika v rámci svazku

2

Stát vydavatele periodika

PL - Polská republika

Počet stran výsledku

21

Strana od-do

119-139

Kód UT WoS článku

000376610400002

EID výsledku v databázi Scopus

2-s2.0-84973517393