论文标题
近端和均匀的近似套件在$ l^p $中
Proximinality and uniformly approximable sets in $L^p$
论文作者
论文摘要
对于[1，\ infty] $中的任何$ p \，我们证明，最多$ k $不同的值的简单功能集为$ l^p $，对于所有$ k \ geq 1 $。我们介绍了$ l^p $的均匀近似子集的类别，该类比均匀集成的集合大。如果$ p \在[1，\ infty）$中的$ p $变量的特征以及如果$ p = \ infty $，则以$ p $变量为特征。我们研究均匀近似集的属性。特别是，我们证明了一个均匀近似界的集合的凸壳也均匀地近似，并且该类别在Hölder变换下是稳定的。我们还证明，对于[1，\ infty）$中的$ p \，$ l^p $的单位球在且仅当$ l^p $是有限维度时，而仅当$ p = \ intty $时，单位球总是均匀近似。
For any $p\in[1,\infty]$, we prove that the set of simple functions taking at most $k$ different values is proximinal in $L^p$ for all $k\geq 1$. We introduce the class of uniformly approximable subsets of $L^p$, which is larger than the class of uniformly integrable sets. This new class is characterized in terms of the $p$-variation if $p\in[1,\infty)$ and in terms of covering numbers if $p=\infty$. We study properties of uniformly approximable sets. In particular, we prove that the convex hull of a uniformly approximable bounded set is also uniformly approximable and that this class is stable under Hölder transformations. We also prove that, for $p\in [1,\infty)$, the unit ball of $L^p$ is uniformly approximable if and only if $L^p$ is finite-dimensional, while for $p=\infty$ the unit ball is always uniformly approximable.
