Abstract
We introduce two measures of weak non-compactness Ja_E and Ja that quantify, via distances, the idea of boundary that lies behind James's Compactness Theorem. These measures tell us, for a bounded subset C of a Banach space E and for given x* is an element of E*, how far from E or C one needs to go to find x** in w*-cl(C) with x**(x*) = sup x*(C).
A quantitative version of James's Compactness Theorem is proved using Ja_E and Ja, and in particular it yields the following result. Let C be a closed convex bounded subset of a Banach space E and r > 0.
If there is an element x_0** in w*-cl(C) whose distance to C is greater than r, then there is x* is an element of E* such that each x** is an element of w*-cl(C) at which sup x*(C) is attained has distance to E greater than 1/2 r. 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.