Biduality and density in Lipschitz function spaces

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 ${\mathrm{lip}}_0(X,d^{\alpha }{)}^{\ast \ast }$ for any $\alpha$ in $(0,1)$.

7 pages