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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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References


Bade, W. G., Curtis, Jr., P. C., and Dales, H. G., Amenability and weak amenability for Beurling and Lipschitz algebras, Proc. London Math. Soc. (3) 55 (1987), no. 2, 359–377. https://doi.org/10.1093/plms/s3-55_2.359

Bierstedt, K. D. and Summers, W. H., Biduals of weighted Banach spaces of analytic functions, J. Austral. Math. Soc. Ser. A 54 (1993), no. 1, 70–79.

Brown, B. M., Elliott, D., and Paget, D. F., Lipschitz constants for the Bernstein polynomials of a Lipschitz continuous function, J. Approx. Theory 49 (1987), no. 2, 196–199. https://doi.org/10.1016/0021-9045(87)90087-6

Ciesielski, Z., On the isomorphisms of the spaces $H_alpha $ and $m$, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 8 (1960), 217–222.

Godefroy, G. and Kalton, N. J., Lipschitz-free Banach spaces, Studia Math. 159 (2003), no. 1, 121–141. https://doi.org/10.4064/sm159-1-6

Jiménez-Vargas, A., Sepulcre, J. M., and Villegas-Vallecillos, M., Lipschitz compact operators, J. Math. Anal. Appl. 415 (2014), no. 2, 889–901. https://doi.org/10.1016/j.jmaa.2014.02.012

Johnson, J. A., Banach spaces of Lipschitz functions and vector-valued Lipschitz functions, Trans. Amer. Math. Soc. 148 (1970), 147–169. https://doi.org/10.2307/1995044

Kalton, N. J., Spaces of Lipschitz and Hölder functions and their applications, Collect. Math. 55 (2004), no. 2, 171–217.

Katznelson, Y., An introduction to harmonic analysis, third ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004. https://doi.org/10.1017/CBO9781139165372

de Leeuw, K., Banach spaces of Lipschitz functions, Studia Math. 21 (1961/1962), 55–66.

Weaver, N., Lipschitz algebras, World Scientific Publishing Co., Inc., River Edge, NJ, 1999. https://doi.org/10.1142/4100




DOI: http://dx.doi.org/10.7146/math.scand.a-25987

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ISSN 0025-5521 (print) ISSN 1903-1807 (online)

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