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Stability of Lipschitz-type functions under pointwise product and reciprocation

Abstract

This article provides necessary and sufficient conditions on the structure of a metric space such that for various vector lattices of real-valued Lipschitz-type functions defined on the metric space, the vector lattice is stable under pointwise product, and such that the reciprocal of each non-vanishing member of the vector lattice remains in the vector lattice. In each case the family of metric spaces for which the first property holds contains the family of metric spaces for which the second property holds. At the end we prove some extension theorems for classes of locally Lipschitz functions that complement known results for Cauchy continuous functions and for uniformly continuous functions.

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Correspondence to Gerald Beer.

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The first author would like to thank Universidad Complutense for its hospitality in September 2019. The second author is partially supported by grants MTM2017-83262-C2-2-P and Fundación Séneca CARM 20906/PI/18. He is also supported by a postdoctoral grant from Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia (Spain). The third author was partially supported by DGES grant PGC2018-097286-B-I00 (Spain).

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Beer, G., García-Lirola, L.C. & Garrido, M.I. Stability of Lipschitz-type functions under pointwise product and reciprocation. RACSAM 114, 120 (2020). https://doi.org/10.1007/s13398-020-00847-x

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Keywords

  • Lipschitz function
  • Locally Lipschitz function
  • Lipschitz in the small function
  • Cauchy-Lipschitz function
  • Pointwise product
  • Reciprocation
  • UC-space
  • Cofinal completeness
  • Modulus of continuity

Mathematics Subject Classification

  • Primary 54E40
  • 26A16
  • Secondary 46E15
  • 54C30