Extensions of Continuous and Lipschitz Functions
Printed: Jun 2000
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 $X$.
extension, continous, Lipschitz, Banach space
54C20 - Extension of maps
46B10 - Duality and reflexivity [See also 46A25]