A quantitative version of James' compactness theorem

Abstract

We introduce two measures of weak non-compactness $J{a}_E$ 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 $x^{\ast }\in E^{\ast }$, how far from $E$ or $C$ one needs to go to find $x^{\ast \ast }\in {\bar{C}}^{w^{\ast }}\subset E^{\ast \ast }$ with $x^{\ast \ast }(x^{\ast })=\sup x^{\ast }(C)$. A quantitative version of James' compactness theorem is proved using $J{a}_E$ 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 $x_0^{\ast \ast }$ in ${\bar{C}}^{w^{\ast }}$ whose distance to $C$ is greater than $r$, then there is $x^{\ast }\in E^{\ast }$ such that each $x^{\ast \ast }\in {\bar{C}}^{w^{\ast }}$ at which $\sup x^{\ast }(C)$ is attained has distance to $E$ greater than $r/2$.} We indeed establish that $J{a}_E$ 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.

16 pages