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.
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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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A. Jiménez-Vargas is supported by the Spanish Ministry of Economy and Competitiveness Project No. MTM2014-58984-P and the European Regional Development Fund (ERDF), and Junta of 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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Jiménez-Vargas, A., Rueda, P. Isometric representations of weighted spaces of little Lipschitz functions. Rev Mat Complut 31, 333–350 (2018). https://doi.org/10.1007/s13163-018-0258-5
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DOI: https://doi.org/10.1007/s13163-018-0258-5