Biduality and density in Lipschitz function spaces
DOI:
https://doi.org/10.7146/math.scand.a-25987Abstract
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 alternative 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 $\alpha \in (0,1)$.
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