论文标题
封闭的线性空间包括强烈的规范获得Lipschitz映射
Closed linear spaces consisting of strongly norm attaining Lipschitz mappings
论文作者
论文摘要
给定一个指定的度量空间$ m $，我们在存在$ \ operatatorName {lip} _0（m）$的$ n $二维线性子空间时进行研究，该子空间由\ mathbb {n} $中的$ n \ in \ mathbb {n} $组成。我们表明，无限度量空间始终是这种情况，为问题提供了确定的答案。我们还研究了此类无限维封闭线性子空间$ y $的可能大小，以及逆问题，即，考虑到存在这样的子空间$ y $的公制空间$ m $的可能尺寸。我们还表明，如果公制空间$ m $是$σ$ - 反理，那么上述子空间$ y $需要始终可分开且异态性多面体，并且我们证明，对于包含$ [0,1] $的空间，它们可以是无限的。
Given a pointed metric space $M$, we study when there exist $n$-dimensional linear subspaces of $\operatorname{Lip}_0(M)$ consisting of strongly norm-attaining Lipschitz functionals, for $n\in\mathbb{N}$. We show that this is always the case for infinite metric spaces, providing a definitive answer to the question. We also study the possible sizes of such infinite-dimensional closed linear subspaces $Y$, as well as the inverse question, that is, the possible sizes of the metric space $M$ given that such a subspace $Y$ exists. We also show that if the metric space $M$ is $σ$-precompact, then the aforementioned subspaces $Y$ need to be always separable and isomorphically polyhedral, and we show that for spaces containing $[0,1]$ isometrically, they can be infinite-dimensional.
