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A note on a Residual Subset of Lipschitz Functions on Metric Spaces

Published online by Cambridge University Press:  30 December 2014

Fabio Cavalletti
Fabio Cavalletti*
Affiliation:
Rheinisch-Westfälische Technische Hochschule Aachen University, Department of Mathematics, Templergraben 64, 52062 Aachen, Germany (cavalletti@instmath.rwth-aachen.de)
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Abstract

Let (X, d) be a quasi-convex, complete and separable metric space with reference probability measure m. We prove that the set of real-valued Lipschitz functions with non-zero pointwise Lipschitz constant m-almost everywhere is residual, and hence dense, in the Banach space of Lipschitz and bounded functions. The result is the metric analogous to a result proved for real-valued Lipschitz maps defined on ℝ2 by Alberti et al.

Keywords

Lipschitz functionsnon-zero gradientresidual setsPrimary 53C2330Lxx
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 58 , Issue 3 , October 2015 , pp. 631 - 636
Copyright
Copyright © Edinburgh Mathematical Society 2015 

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References

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