Dunford–Pettis type properties and the Grothendieck property for function spaces

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F20%3A00532217" target="_blank" >RIV/67985840:_____/20:00532217 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.1007/s13163-019-00336-9" target="_blank" >https://doi.org/10.1007/s13163-019-00336-9</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s13163-019-00336-9" target="_blank" >10.1007/s13163-019-00336-9</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Dunford–Pettis type properties and the Grothendieck property for function spaces

Popis výsledku v původním jazyce

For a Tychonoff space X, let Ck(X) and Cp(X) be the spaces of real-valued continuous functions C(X) on X endowed with the compact-open topology and the pointwise topology, respectively. If X is compact, the classic result of A. Grothendieck states that Ck(X) has the Dunford–Pettis property and the sequential Dunford–Pettis property. We extend Grothendieck’s result by showing that Ck(X) has both the Dunford–Pettis property and the sequential Dunford–Pettis property if X satisfies one of the following conditions: (1) X is a hemicompact space, (2) X is a cosmic space (= a continuous image of a separable metrizable space), (3) X is the ordinal space [0 , κ) for some ordinal κ, or (4) X is a locally compact paracompact space. We show that if X is a cosmic space, then Ck(X) has the Grothendieck property if and only if every functionally bounded subset of X is finite. We prove that Cp(X) has the Dunford–Pettis property and the sequential Dunford–Pettis property for every Tychonoff space X, and Cp(X) has the Grothendieck property if and only if every functionally bounded subset of X is finite.

Název v anglickém jazyce

Dunford–Pettis type properties and the Grothendieck property for function spaces

Popis výsledku anglicky

For a Tychonoff space X, let Ck(X) and Cp(X) be the spaces of real-valued continuous functions C(X) on X endowed with the compact-open topology and the pointwise topology, respectively. If X is compact, the classic result of A. Grothendieck states that Ck(X) has the Dunford–Pettis property and the sequential Dunford–Pettis property. We extend Grothendieck’s result by showing that Ck(X) has both the Dunford–Pettis property and the sequential Dunford–Pettis property if X satisfies one of the following conditions: (1) X is a hemicompact space, (2) X is a cosmic space (= a continuous image of a separable metrizable space), (3) X is the ordinal space [0 , κ) for some ordinal κ, or (4) X is a locally compact paracompact space. We show that if X is a cosmic space, then Ck(X) has the Grothendieck property if and only if every functionally bounded subset of X is finite. We prove that Cp(X) has the Dunford–Pettis property and the sequential Dunford–Pettis property for every Tychonoff space X, and Cp(X) has the Grothendieck property if and only if every functionally bounded subset of X is finite.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2020

Kód důvěrnosti údajů

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

Název periodika

Revista Mathématica Complutense

ISSN

1139-1138

e-ISSN

—

Svazek periodika

33

Číslo periodika v rámci svazku

3

Stát vydavatele periodika

ES - Španělské království

Počet stran výsledku

14

Strana od-do

871-884

Kód UT WoS článku

000567467200011

EID výsledku v databázi Scopus

2-s2.0-85076085610