  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 $(0,1)$.
