Polish spaces of Banach spaces: Complexity of isometry and isomorphism classes

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F24%3A00599029" target="_blank" >RIV/67985840:_____/24:00599029 - isvavai.cz</a>

Nalezeny alternativní kódy

RIV/00216208:11320/24:10492949

Výsledek na webu

<a href="https://doi.org/10.1017/S1474748023000440" target="_blank" >https://doi.org/10.1017/S1474748023000440</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1017/S1474748023000440" target="_blank" >10.1017/S1474748023000440</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Polish spaces of Banach spaces: Complexity of isometry and isomorphism classes

Popis výsledku v původním jazyce

We study the complexities of isometry and isomorphism classes of separable Banach spaces in the Polish spaces of Banach spaces, recently introduced and investigated by the authors in [14]. We obtain sharp results concerning the most classical separable Banach spaces. We prove that the infinite-dimensional separable Hilbert space is characterized as the unique separable infinite-dimensional Banach space whose isometry class is closed, and also as the unique separable infinitedimensional Banach space whose isomorphism class is Fσ. For p ∈ [1,2) (2,∞), we show that the isometry classes of Lp[0,1] and ℓp are G-complete sets and Fσ-complete sets, respectively. Then we show that the isometry class of c0 is an Fσ-complete set. Additionally, we compute the complexities of many other natural classes of separable Banach spaces, for instance, the class of separable Lp,λ+-spaces, for p,λ ≥ 1, is shown to be a G-set, the class of superreflexive spaces is shown to be an Fσ-set, and the class of spaces with local Π-basis structure is shown to be a Σ06 -set. The paper is concluded with many open problems and suggestions for a future research.

Název v anglickém jazyce

Polish spaces of Banach spaces: Complexity of isometry and isomorphism classes

Popis výsledku anglicky

We study the complexities of isometry and isomorphism classes of separable Banach spaces in the Polish spaces of Banach spaces, recently introduced and investigated by the authors in [14]. We obtain sharp results concerning the most classical separable Banach spaces. We prove that the infinite-dimensional separable Hilbert space is characterized as the unique separable infinite-dimensional Banach space whose isometry class is closed, and also as the unique separable infinitedimensional Banach space whose isomorphism class is Fσ. For p ∈ [1,2) (2,∞), we show that the isometry classes of Lp[0,1] and ℓp are G-complete sets and Fσ-complete sets, respectively. Then we show that the isometry class of c0 is an Fσ-complete set. Additionally, we compute the complexities of many other natural classes of separable Banach spaces, for instance, the class of separable Lp,λ+-spaces, for p,λ ≥ 1, is shown to be a G-set, the class of superreflexive spaces is shown to be an Fσ-set, and the class of spaces with local Π-basis structure is shown to be a Σ06 -set. The paper is concluded with many open problems and suggestions for a future research.

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

2024

Kód důvěrnosti údajů

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

Název periodika

Journal of the Institute of Mathematics of Jussieu

ISSN

1474-7480

e-ISSN

1475-3030

Svazek periodika

23

Číslo periodika v rámci svazku

4

Stát vydavatele periodika

GB - Spojené království Velké Británie a Severního Irska

Počet stran výsledku

39

Strana od-do

1919-1957

Kód UT WoS článku

001112779800001

EID výsledku v databázi Scopus

2-s2.0-85179087321