Smooth and polyhedral norms via fundamental biorthogonal systems

TL;DR

This paper demonstrates that dense subspaces of Banach spaces with fundamental biorthogonal systems can be equipped with smooth and polyhedral norms, extending many classical results and covering a wide class of spaces.

Contribution

It proves the existence of smooth and polyhedral norms on dense subspaces of Banach spaces with fundamental biorthogonal systems, generalizing previous results and covering numerous important classes of Banach spaces.

Findings

Existence of $C^ abla$-smooth norms on dense subspaces

Construction of polyhedral norms with local finite dependence

Application to a broad class of Banach spaces including WLD, $L_1$, and $C(K)$ spaces

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 LUR 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 WLD 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 {\sf MM}, all Banach spaces of density $\omega_1$ are covered by our result.