Lipschitz空间的非紧性和非完备性
Noncompactness and noncompleteness in isometries of Lipschitz spaces
摘要：
对于严格凸赋范空间$E$和$F$以及度量空间$X$和$Y$，我们解决了以下三个关于Lipschitz函数空间$\mathrm{Lip}(X,E)$和$\mathrm{Lip}(Y,F)$之间满射线性等距的问题：\begin{enumerate} \item Characterize those base spaces $X$ and $Y$ for which all isometries are weighted composition maps. \item Give a condition independent of base spaces under which all isometries are weighted composition maps. \item Provide the general form of an isometry, both when it is a weighted composition map and when it is not. \end{enumerate}特别地，我们证明了当$E$和$F$不完备时，对$X$和$Y$的完备性要求是不必要的，这与标量上下文中已知的结果形成鲜明对比。
英文摘要：
We solve the following three questions concerning surjective linear isometries between spaces of Lipschitz functions $\mathrm{Lip}(X,E)$ and $\mathrm{Lip}(Y,F)$, for strictly convex normed spaces $E$ and $F$ and metric spaces $X$ and $Y$: \begin{enumerate} \item Characterize those base spaces $X$ and $Y$ for which all isometries are weighted composition maps. \item Give a condition independent of base spaces under which all isometries are weighted composition maps. \item Provide the general form of an isometry, both when it is a weighted composition map and when it is not. \end{enumerate} In particular, we prove that requirements of completeness on $X$ and $Y$ are not necessary when $E$ and $F$ are not complete, which is in sharp contrast with results known in the scalar context.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
