A Proper Closed Subspace of the Lipschitz Dual Containing the Linear Dual

TL;DR

This paper explores the dual space structure of Lipschitz function spaces, identifying a proper closed subspace that contains the linear dual and establishing algebraic properties.

Contribution

It introduces a specific closed subspace of Lipschitz functions that contains the linear dual and demonstrates its dual space structure and algebraic properties.

Findings

$Lip_0^{ph}(X)$ is a dual space and preannihilator of a closed subspace

The quotient $Lip_0(X)/Lip_0^{ph}(X)$ is a dual space

$(Lip_0^{ph}(X), Lip( ext{·}))$ forms a Banach algebra

Abstract

Motivated by classical results of Lindenstrauss and recent developments by Karn and Mandal, we investigate quotient spaces of the form $Lip_0(X)/\mathcal{A}$, where $\mathcal{A}$ is a finite-dimensional subspace, showing that these quotients are dual spaces with explicitly describable preduals. We then focus on $Lip_0^{ph}(X)$, the space of positively homogeneous real-valued Lipschitz functions. This space satisfies $ X^* \subsetneq Lip_0^{ph}(X) \subsetneq Lip_0(X), $ and is shown to be both a dual space and the preannihilator of a closed subspace of the Lipschitz-free space. Consequently it follows that $\bigslant{Lip_0(X)}{Lip_0^{ph}(X)}$ is also a dual space. Furthermore, with a suitable multiplication, $(Lip_0^{ph}(X), Lip(\cdot))$ forms a Banach algebra, exhibiting structural advantages over $Lip_0(X)$.