A quantitative version of James's Compactness Theorem

Abstract

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 * ∈ E * , how far from E or C one needs to go to find x ** ∈ $\overline{C}^{w^*}$ ⊂ E ** 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 $\overline{C}^{w^*}$ whose distance to C is greater than r, then there is x * ∈ E * such that each x ** ∈ $\overline{C}^{w^*}$ at which sup x * (C) is attained has distance to E greater than ½r . 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.