Smooth norms in dense subspaces of $\ell_p(Γ)$ and operator ranges

Dantas, Sheldon; Hájek, Petr; Russo, Tommaso

doi:10.48550/arxiv.2205.11282

Cited by 1 publication

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“…Several months after the present research was completed, the authors obtained the following result [10], related to Theorem A. If 1 p < ∞, the dense subspace Y p := 0<q<p ℓ q (Γ) of ℓ p (Γ) admits a C ∞ -smooth and LFC norm.…”

Section: Introductionmentioning

confidence: 86%

Smooth and Polyhedral Norms via Fundamental Biorthogonal Systems

Dantas

Hájek

Russo

2022

International Mathematics Research Notices

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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.

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Section: Introductionmentioning

confidence: 86%