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Extensions of Continuous and Lipschitz Functions

Published:2000-06-01
Printed: Jun 2000
• Eva Matoušková
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Abstract

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: extension, continous, Lipschitz, Banach space
 MSC Classifications: 54C20 - Extension of maps 46B10 - Duality and reflexivity [See also 46A25]