James的compactness定理的一个定量形式
A quantitative version of James' compactness theorem
摘要：
我们引入了两种弱非紧性测度$Ja_E$和$Ja$，它们通过距离量化了James紧性定理背后的边界概念。这些测度告诉我们，对于Banach空间$E$的有界子集$C$和给定的$x^*\in E^*$，需要从$E$或$C$走多远才能用$x^{**}(x^*)=\sup x^* (C)$找到$x^{**}\in \bar{C}^{w^*}\subset E^{**}$。利用$Ja_E$和$Ja$证明了James紧性定理的一个定量形式，特别地，它得到了以下结果：{ \it}设$C$是Banach空间的闭凸有界子集$E$和$r>0$ .如果在$\bar{C}^{w^*}$中有一个元素$x_0^{**}$，其到$C$的距离大于$r$，则存在$x^*\in E^*$，使得在其处获得$\sup x^*(C)$的每个$x^{**}\in\bar{C}^{w^*}$到$E$的距离大于$r/2$。}我们确实建立了$Ja_E$和$Ja$是等价于其他措施的弱非紧性研究的文献。我们还收集了特殊的情况和例子表明，当弱非紧性的不同措施之间的不平等可以是平等的，当不平等是尖锐的。
英文摘要：
We introduce two measures of weak non-compactness $Ja_E$ and $Ja$ that quantify, via distances, the idea of boundary behind James' compactness theorem. These measures tell us, for a bounded subset $C$ of a Banach space $E$ and for given $x^*\in E^*$, how far from $E$ or $C$ one needs to go to find $x^{**}\in \bar{C}^{w^*}\subset E^{**}$ with $x^{**}(x^*)=\sup x^* (C)$. A quantitative version of James' compactness theorem is proved using $Ja_E$ and $Ja$, and in particular it yields the following result: {\it Let $C$ be a closed convex bounded subset of a Banach space $E$ and $r>0$. If there is an element $x_0^{**}$ in $\bar{C}^{w^*}$ whose distance to $C$ is greater than $r$, then there is $x^*\in E^*$ such that each $x^{**}\in\bar{C}^{w^*}$ at which $\sup x^*(C)$ is attained has distance to $E$ greater than $r/2$.} We indeed establish that $Ja_E$ and $Ja$ are equivalent to other measures of weak non-compactness studied in the literature. We also collect particular cases and examples showing when the inequalities between the different measures of weak non-compactness can be equalities and when the inequalities are sharp.
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。
