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Extensions of Continuous and Lipschitz Functions

Published online by Cambridge University Press:  20 November 2018

Eva Matoušková
Eva Matoušková*
Affiliation:
Mathematical Institute Czech Academy of Sciences Žitná 25 CZ-11567 Prague Czech Republic, e-mail: matouse@matsrv.math.cas.cz
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Abstract

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We show a result slightly more general than the following. Let $K$ be a compact Hausdorff space, $F$ a closed subset of $K$ , and $d$ a lower semi-continuous metric on $K$ . Then each continuous function $f$ on $F$ which is Lipschitz in $d$ admits a continuous extension on $K$ which is Lipschitz in $d$ . The extension has the same supremum norm and the same Lipschitz constant.

As a corollary we get that a Banach space $X$ is reflexive if and only if each bounded, weakly continuous and norm Lipschitz function defined on a weakly closed subset of $X$ admits a weakly continuous, norm Lipschitz extension defined on the entire space $X$ .

Keywords

54C2046B10extensioncontinuousLipschitzBanach space

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 43 , Issue 2 , 01 June 2000, pp. 208 - 217
Copyright
Copyright © Canadian Mathematical Society 2000

References

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