Extensions of Continuous and Lipschitz Functions

Canadian mathematical bulletin

关键词: extension;    continous;    Lipschitz;    Banach space;   

DOI  :  10.4153/CMB-2000-028-0

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

We show a result slightly more general than the following. Let $K$be a compact Hausdorff space, $F$ a closed subset of $K$, and $d$ alower 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 supremumnorm and the same Lipschitz constant. As a corollary we get that a Banach space $X$ is reflexive if and onlyif each bounded, weakly continuous and norm Lipschitz functiondefined on a weakly closed subset of $X$ admits a weakly continuous,norm Lipschitz extension defined on the entire space $X$.

【 授权许可】

Unknown