Biduality and density in Lipschitz function spaces

  • A. Jiménez-Vargas
  • J. M. Sepulcre
  • M. Villegas-Vallecillos

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 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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Published
2017-09-22
How to Cite
Jiménez-Vargas, A., Sepulcre, J., & Villegas-Vallecillos, M. (2017). Biduality and density in Lipschitz function spaces. MATHEMATICA SCANDINAVICA, 121(1), 92-100. https://doi.org/10.7146/math.scand.a-25987
Section
Articles