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

Motivated by classical results of Lindenstrauss and recent developments by Karn and Mandal, we investigate quotient spaces of the form $Li{p}_0(X)\mathrm{/}\mathcal{A}$, where $\mathcal{A}$ is a finite-dimensional subspace, showing that these quotients are dual spaces with explicitly describable preduals. We then focus on $Li{p}_0^{ph}(X)$, the space of positively homogeneous real-valued Lipschitz functions. This space satisfies $X^{\ast }⊊Li{p}_0^{ph}(X)⊊Li{p}_0(X),$ 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 $\text{bigslant}Li{p}_0(X)Li{p}_0^{ph}(X)$ is also a dual space. Furthermore, with a suitable multiplication, $(Li{p}_0^{ph}(X),Lip(\cdot ))$ forms a Banach algebra, exhibiting structural advantages over $Li{p}_0(X)$.