Smooth and polyhedral norms via fundamental biorthogonal systems

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F23%3A00575126" target="_blank" >RIV/67985840:_____/23:00575126 - isvavai.cz</a>

Nalezeny alternativní kódy

RIV/68407700:21230/23:00370646

Výsledek na webu

<a href="https://doi.org/10.1093/imrn/rnac211" target="_blank" >https://doi.org/10.1093/imrn/rnac211</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1093/imrn/rnac211" target="_blank" >10.1093/imrn/rnac211</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Smooth and polyhedral norms via fundamental biorthogonal systems

Popis výsledku v původním jazyce

Let X be a Banach space with a fundamental biorthogonal system, and let y be the dense subspace spanned by the vectors of the system. We prove that y admits a C-infinity-smooth norm that locally depends on finitely many coordinates (LFC, for short), as well as a polyhedral norm that locally depends on finitely many coordinates. As a consequence, we also prove that y admits locally finite, sigma-uniformly discrete C-infinity-smooth and LFC partitions of unity and a C-1-smooth locally uniformly rotund norm. This theorem substantially generalises several results present in the literature and gives a complete picture concerning smoothness in such dense subspaces. Our result covers, for instance, every weakly Lindelof determined Banach space (hence, all reflexive ones), L-1 (mu) for every measure mu, l(infinity) (Gamma) spaces for every set Gamma, C(K) spaces where K is a Valdivia compactum or a compact Abelian group, duals of Asplund spaces, or preduals of Von Neumann algebras. Additionally, under Martin Maximum MM, all Banach spaces of density omega(1) are covered by our result.

Název v anglickém jazyce

Smooth and polyhedral norms via fundamental biorthogonal systems

Popis výsledku anglicky

Let X be a Banach space with a fundamental biorthogonal system, and let y be the dense subspace spanned by the vectors of the system. We prove that y admits a C-infinity-smooth norm that locally depends on finitely many coordinates (LFC, for short), as well as a polyhedral norm that locally depends on finitely many coordinates. As a consequence, we also prove that y admits locally finite, sigma-uniformly discrete C-infinity-smooth and LFC partitions of unity and a C-1-smooth locally uniformly rotund norm. This theorem substantially generalises several results present in the literature and gives a complete picture concerning smoothness in such dense subspaces. Our result covers, for instance, every weakly Lindelof determined Banach space (hence, all reflexive ones), L-1 (mu) for every measure mu, l(infinity) (Gamma) spaces for every set Gamma, C(K) spaces where K is a Valdivia compactum or a compact Abelian group, duals of Asplund spaces, or preduals of Von Neumann algebras. Additionally, under Martin Maximum MM, all Banach spaces of density omega(1) are covered by our result.

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

2023

Kód důvěrnosti údajů

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

Název periodika

International Mathematics Research Notices

ISSN

1073-7928

e-ISSN

1687-0247

Svazek periodika

2023

Číslo periodika v rámci svazku

16

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

31

Strana od-do

13909-13939

Kód UT WoS článku

000836338400001

EID výsledku v databázi Scopus

2-s2.0-85168586630