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 $(0,1)$.
Categories: math.FA