Smooth and polyhedral norms via fundamental biorthogonal systems

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.

Int. Math. Res. Not. IMRN (online first)

References in corpus (2)

• Smooth norms in dense subspaces of Banach spaces
• On densely isomorphic normed spaces

Cited by in corpus (2)

• Rotund Gateaux smooth norms which are not locally uniformly rotund
• Smooth norms in dense subspaces of ${\ell }_p(\Gamma )$ and operator ranges