Smooth and polyhedral norms via fundamental biorthogonal systems
Result description
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.
Keywords
Frechet differentiable normsMazur intersection propertyBanach spaces
The result's identifiers
Result code in IS VaVaI
Alternative codes found
RIV/68407700:21230/23:00370646
Result on the web
DOI - Digital Object Identifier
Alternative languages
Result language
angličtina
Original language name
Smooth and polyhedral norms via fundamental biorthogonal systems
Original language description
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.
Czech name
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Czech description
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Classification
Type
Jimp - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
—
OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
Result was created during the realization of more than one project. More information in the Projects tab.
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2023
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
International Mathematics Research Notices
ISSN
1073-7928
e-ISSN
1687-0247
Volume of the periodical
2023
Issue of the periodical within the volume
16
Country of publishing house
US - UNITED STATES
Number of pages
31
Pages from-to
13909-13939
UT code for WoS article
000836338400001
EID of the result in the Scopus database
2-s2.0-85168586630
Basic information
Result type
Jimp - Article in a specialist periodical, which is included in the Web of Science database
OECD FORD
Pure mathematics
Year of implementation
2023