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Functions that are Lipschitz in the small

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Abstract

Let X and Y be metric spaces. A function \(f:X\rightarrow Y\) is said to be Lipschitz in the small if there are \(r> 0\) and \(K<\infty \) so that \(d(f(u),f(v)) \le Kd(u,v)\) for any \(u,v \in X\) with \(d(u,v) \le r\). We find necessary and sufficient conditions on a subset A of X such that \(f_{|A}\) is Lipschitz for every function f that is Lipschitz in the small on X. We also find necessary and sufficient conditions on X for \({\text {*}}{LS}\left( X\right) \) to be linearly order isomorphic to \({\text {Lip}}(Y)\) for some metric space Y.

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References

  1. Luukkainen, J.: Rings of functions in Lipschitz topology. Ann. Acad. Sci. Fenn. Ser. A. I. Math. 4, 119–135 (1978–1979)

  2. Beer, Gerald, Garrido, M.Isabel: Bornologies and locally Lipschitz functions. Bull. Aust. Math. Soc. 90, 257–263 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Beer, Gerald, Garrido, M.Isabel: Locally Lipschitz functions, cofinal completeness, and UC spaces. J. Math. Anal. Appl. 428, 804–816 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Garrido, M.Isabel, Jaramillo, Jesús A.: Lipschitz-type functions on metric spaces. J. Math. Anal. Appl. 340, 282–290 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Beer, Gerald, Garrido, M.Isabel: On the uniform approximation of Cauchy continuous functions. Topol. Appl. 208, 1–9 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Anthony, G.: O’Farrell, when uniformly-continuous implies bounded. Irish Math. Soc. Bull. 53, 53–56 (2004)

  7. Garrido, M.Isabel, Meroño, Ana S.: Two classes of metric spaces. Appl. Gen. Topol 17, 57–70 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Leung Denny H., Tang, W.K.: Nonlinear order isomorphisms on function spaces. Diss. Math. (to appear)

  9. Michael, E.: A short proof of the Arens–Eells theorem. Proc. Am. Math. Soc. 15, 415–416 (1964)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors thank two anonymous referees for their helpful comments.

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Authors and Affiliations

  1. Department of Mathematics, National University of Singapore, Singapore, 119076, Singapore

    Denny H. Leung

  2. Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, 637371, Singapore

    Wee-Kee Tang

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  1. Denny H. Leung
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  2. Wee-Kee Tang
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Correspondence to Wee-Kee Tang.

Additional information

Research of the W.-K. Tang is partially supported by AcRF project no. RG26/14.

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Leung, D.H., Tang, WK. Functions that are Lipschitz in the small. Rev Mat Complut 30, 25–34 (2017). https://doi.org/10.1007/s13163-016-0205-2

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  • DOI: https://doi.org/10.1007/s13163-016-0205-2

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