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- Title
Smooth and Polyhedral Norms via Fundamental Biorthogonal Systems.
- Authors
Dantas, Sheldon; Hájek, Petr; Russo, Tommaso
- Abstract
Let |$\mathcal {X}$| be a Banach space with a fundamental biorthogonal system, and let |$\mathcal {Y}$| be the dense subspace spanned by the vectors of the system. We prove that |$\mathcal {Y}$| admits a |$C^\infty $| -smooth norm that locally depends on finitely many coordinates (LFC, for short), as well as a polyhedral norm that locally depends on finitely many coordinates. As a consequence, we also prove that |$\mathcal {Y}$| admits locally finite, |$\sigma $| -uniformly discrete |$C^\infty $| -smooth and LFC partitions of unity and a |$C^1$| -smooth locally uniformly rotund norm. This theorem substantially generalises several results present in the literature and gives a complete picture concerning smoothness in such dense subspaces. Our result covers, for instance, every weakly Lindelöf determined Banach space (hence, all reflexive ones), |$L_1(\mu)$| for every measure |$\mu $| , |$\ell _\infty (\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.
- Subjects
VALDIVIA (Chile); BIORTHOGONAL systems; VON Neumann, John, 1903-1957; BANACH spaces; ABELIAN groups; COMPACT groups; POLYHEDRAL functions; VON Neumann algebras
- Publication
IMRN: International Mathematics Research Notices, 2023, Vol 2023, Issue 16, p13909
- ISSN
1073-7928
- Publication type
Article
- DOI
10.1093/imrn/rnac211