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

TL;DR

This paper constructs dense subspaces of ll_p(\u03b3) with smooth norms that approximate the standard norm, contain dense operator ranges, and are not spanned by biorthogonal systems, revealing new geometric properties.

Contribution

It introduces dense subspaces of ll_p(\u03b3) with smooth norms that approximate the ll_p norm and contain dense operator ranges, expanding understanding of geometric structures in Banach spaces.

Findings

Existence of smooth norms on dense subspaces approximating ll_p norms.

Construction of dense subspaces with dense operator ranges.

Non-existence of smooth norms on certain non-separable operator ranges.

Abstract

For $1\leq p<\infty$, we prove that the dense subspace $\mathcal{Y}_p$ of $\ell_p(\Gamma)$ comprising all elements $y$ such that $y \in \ell_q(\Gamma)$ for some $q \in (0,p)$ admits a $C^{\infty}$-smooth norm which locally depends on finitely many coordinates. Moreover, such a norm can be chosen as to approximate the $\left\Vert\cdot \right\Vert_p $-norm. This provides examples of dense subspaces of $\ell_p(\Gamma)$ with a smooth norm which have the maximal possible linear dimension and are not obtained as the linear span of a biorthogonal system. Moreover, when $p>1$ or $\Gamma$ is countable, such subspaces additionally contain dense operator ranges; on the other hand, no non-separable operator range in $\ell_1(\Gamma)$ admits a $C^1$-smooth norm.