This is an announcement for the paper "A quantitative version of James' compactness theorem" by Bernardo Cascales, Ondrej F.K. Kalenda and Jiri Spurny.

Abstract: 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 $x^{**}\in \overline{C}^{w^*}\subset E^{**}$ 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 $\overline{C}^{w^*}$ whose distance to $C$ is greater than $r$, then there is $x^*\in E^*$ such that each $x^{**}\in\overline{C}^{w^*}$ 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.

Archive classification: math.FA

Remarks: 16 pages

Submitted from: kalenda@karlin.mff.cuni.cz

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1005.5693

or

http://arXiv.org/abs/1005.5693