Biduality and density in Lipschitz function spaces

TL;DR

This paper investigates when Lipschitz function spaces are biduals of little Lipschitz spaces, establishing a density criterion that simplifies classical isometric isomorphism results for fractional metrics.

Contribution

It introduces a density-based criterion for biduality in Lipschitz spaces and provides an alternative proof for classical fractional metric cases.

Findings

Density of the unit ball of little Lipschitz functions implies biduality.

Established a new criterion for biduality in Lipschitz function spaces.

Provided an alternative proof for classical fractional metric results.

Abstract

For pointed compact metric spaces $(X,d)$, we address the biduality problem as to when the space of Lipschitz functions $\mathrm{Lip}_0(X,d)$ is isometrically isomorphic to the bidual of the space of little Lipschitz functions $\mathrm{lip}_0(X,d)$, and show that this is the case whenever the closed unit ball of $\mathrm{lip}_0(X,d)$ is dense in the closed unit ball of $\mathrm{Lip}_0(X,d)$ with respect to the topology of pointwise convergence. Then we apply our density criterion to prove in an alternate way the real version of a classical result which asserts that $\mathrm{Lip}_0(X,d^\alpha)$ is isometrically isomorphic to $\mathrm{lip}_0(X,d^\alpha)^{**}$ for any $\alpha$ in $(0,1)$.