Separated sets and Auerbach systems in Banach spaces

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F20%3A00532935" target="_blank" >RIV/67985840:_____/20:00532935 - isvavai.cz</a>

Nalezeny alternativní kódy

RIV/68407700:21230/20:00346253

Výsledek na webu

<a href="https://doi.org/10.1090/tran/8160" target="_blank" >https://doi.org/10.1090/tran/8160</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1090/tran/8160" target="_blank" >10.1090/tran/8160</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Separated sets and Auerbach systems in Banach spaces

Popis výsledku v původním jazyce

The paper elucidates the relationship between the density of a Banach space and possible sizes of Auerbach systems and well-separated subsets of its unit sphere. For example, it is proved that for a large enough space $ X$, the unit sphere $ S_X$ always contains an uncountable $ (1+)$-separated subset. In order to achieve this, new results concerning the existence of large Auerbach systems are established, that happen to be sharp for the class of weakly Lindelöf determined (WLD) spaces. In fact, we offer the first consistent example of a non-separable WLD Banach space that contains no uncountable Auerbach system, as witnessed by a renorming of $ c_0(omega _1)$. Moreover, the following optimal results for the classes of, respectively, reflexive and super-reflexive spaces are established: the unit sphere of an infinite-dimensional reflexive space contains a symmetrically $ (1+varepsilon )$-separated subset of any regular cardinality not exceeding the density of $ X$, should the space $ X$ be super-reflexive, the unit sphere of $ X$ contains such a subset of cardinality equal to the density of $ X$. The said problem is studied for other classes of spaces too, including WLD spaces, RNP spaces, or strictly convex ones.

Název v anglickém jazyce

Separated sets and Auerbach systems in Banach spaces

Popis výsledku anglicky

The paper elucidates the relationship between the density of a Banach space and possible sizes of Auerbach systems and well-separated subsets of its unit sphere. For example, it is proved that for a large enough space $ X$, the unit sphere $ S_X$ always contains an uncountable $ (1+)$-separated subset. In order to achieve this, new results concerning the existence of large Auerbach systems are established, that happen to be sharp for the class of weakly Lindelöf determined (WLD) spaces. In fact, we offer the first consistent example of a non-separable WLD Banach space that contains no uncountable Auerbach system, as witnessed by a renorming of $ c_0(omega _1)$. Moreover, the following optimal results for the classes of, respectively, reflexive and super-reflexive spaces are established: the unit sphere of an infinite-dimensional reflexive space contains a symmetrically $ (1+varepsilon )$-separated subset of any regular cardinality not exceeding the density of $ X$, should the space $ X$ be super-reflexive, the unit sphere of $ X$ contains such a subset of cardinality equal to the density of $ X$. The said problem is studied for other classes of spaces too, including WLD spaces, RNP spaces, or strictly convex ones.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

Výsledek vznikl pri realizaci vícero projektů. Více informací v záložce Projekty.

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

American Mathematical Society. Transactions

ISSN

0002-9947

e-ISSN

—

Svazek periodika

373

Číslo periodika v rámci svazku

10

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

38

Strana od-do

6961-6998

Kód UT WoS článku

000576760300007

EID výsledku v databázi Scopus

2-s2.0-85085767164