Smooth and polyhedral norms via fundamental biorthogonal systems

0575126 - MÚ 2024 RIV US eng J - Journal Article
Dantas, S. - Hájek, P. - Russo, Tommaso
Smooth and polyhedral norms via fundamental biorthogonal systems.
International Mathematics Research Notices. Roč. 2023, č. 16 (2023), s. 13909-13939. ISSN 1073-7928. E-ISSN 1687-0247
R&D Projects: GA ČR(CZ) GF20-22230L
Institutional support: RVO:67985840
Keywords : Frechet differentiable norms * Mazur intersection property * Banach spaces
OECD category: Pure mathematics
Impact factor: 1, year: 2022
Method of publishing: Limited access
https://doi.org/10.1093/imrn/rnac211

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.
Permanent Link: https://hdl.handle.net/11104/0344980