A Proper Closed Subspace of the Lipschitz Dual Containing the Linear Dual
Arindam Mandal
TL;DR
The paper studies duality structures arising from quotients of $Lip_0(X)$ by finite-dimensional subspaces and introduces $Lip_0^{ph}(X)$, the space of positively homogeneous Lipschitz maps, as a proper closed subspace between $X^*$ and $Lip_0(X)$. It shows $Lip_0(X)/Lip_0^{ph}(X)$ is a dual space with an explicit predual, and $Lip_0^{ph}(X)$ itself is the pre-annihilator of a closed subspace of the Lipschitz-free space $F(X)$, enabling a dual description of the quotient. Furthermore, $Lip_0^{ph}(X)$ is endowed with a Banach-algebra structure via a modified multiplication, and a McShane-type extension theorem is established for positive homogeneous Lipschitz maps. The work also proves that $F^{ph}(X)$ complements inside $F(X)\oplus \mathbb{R}$ and contains $L^1(\mathbb{R})$ when $\dim X\ge 2$, implying $Lip_0^{ph}(X)$ is infinite-dimensional in multi-dimensional settings. These results collectively highlight the structural advantages of $Lip_0^{ph}(X)$ over $Lip_0(X)$ in the Lipschitz-dual framework.
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 $Lip_0(X)$.