Revista Matemática Complutense

, Volume 31, Issue 2, pp 333–350 | Cite as

Isometric representations of weighted spaces of little Lipschitz functions

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Abstract

Given a compact pointed metric space X and a weight v on the complement of the diagonal set in \(X\times X\), we prove that the Banach space \(\mathrm {lip}_v(X)\) of all weighted little Lipschitz scalar-valued functions on X vanishing at the basepoint, equipped with the weighted Lipschitz norm, embeds almost isometrically into \(c_0\). This result has many consequences on the structure of those Banach spaces and their duals. Moreover, we prove that this isomorphism can never be an isometric embedding whenever X is a \(\mathbb {T}\)-balanced subset containing 0 and compact for some metrizable topology of a complex Banach space and v is a radial 0-weight.

Keywords

Lipschitz function Little Lipschitz function Weighted Banach space 

Mathematics Subject Classification

46E15 46A20 

Notes

Acknowledgements

The authors are very grateful to the two reviewers of the paper for many helpful comments and some corrections. A. Jiménez-Vargas is supported by Ministerio de Economía y Competitividad and FEDER project no. MTM2014-58984-P and Junta de Andalucía grant FQM-194. P. Rueda is supported by Ministerio de Economía y Competitividad and FEDER under project MTM2016-77054-C2-1-P. This work was done while P. Rueda was visiting the Department of Mathematical Sciences at Kent State University supported by Ministerio de Educación, Cultura y Deporte PRX16/00037. She thanks this Department for its kind hospitality.

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Copyright information

© Universidad Complutense de Madrid 2018

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad de AlmeríaAlmeríaSpain
  2. 2.Departamento de Análisis MatemáticoUniversidad de ValenciaBurjassotSpain

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