Smooth and Polyhedral Norms via Fundamental Biorthogonal Systems
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Title
Smooth and Polyhedral Norms via Fundamental Biorthogonal SystemsDate
2022-08-04Publisher
Oxford University PressBibliographic citation
Dantas, Sheldon, Hájek, Petr, Russo, Tommaso. Smooth and Polyhedral Norms via Fundamental Biorthogonal Systems, International Mathematics Research Notices, Volume 2023, Issue 16, August 2023, Pages 13909–13939Type
info:eu-repo/semantics/articlePublisher version
https://academic.oup.com/imrn/article/2023/16/13909/6655610?login=trueVersion
info:eu-repo/semantics/submittedVersionSubject
Abstract
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∞-smooth norm that locally depends on finitely many ... [+]
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∞-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, σ-uniformly discrete C∞-smooth and LFC partitions of unity and a C1-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 Lindelöf determined Banach space (hence, all reflexive ones), L1(μ) for every measure μ, ℓ∞(Γ) spaces for every set Γ, 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 ω1 are covered by our result. [-]
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International Mathematics Research Notices, Volume 2023, Issue 16, August 2023Funder Name
Agencia Estatal de Investigación (AEI) | Grantová Agentura České Republiky (GAČR) | Istituto Nazionale di Alta Matematica (INdAM)
Project code
PID2019-106529GB-I00/AEI/10.13039/501100011033 | PID2021-122126NB-C33/MCIN/AEI/10.13039/501100011033 | 20-22230L | RVO (67985840)
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- IMAC_Articles [117]