Proceedings of the Edinburgh Mathematical Society (Series 2)

Table of Contents - June 2012 - Volume 55, Issue 02  

Research Article

A quantitative version of James's Compactness Theorem

Bernardo Cascalesa1, Ondřej F. K. Kalendaa2 and Jiří Spurnýa2

a1 Departamento de Matemáticas, Universidad de Murcia, 30100 Espinardo, Murcia, Spain (beca@um.es)

a2 Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 18675 Praha 8, Czech Republic (kalenda@karlin.mff.cuni.cz; spurny@karlin.mff.cuni.cz)

Abstract

We introduce two measures of weak non-compactness JaE 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 JaE 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 JaE 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.

(Received May 31 2010)

Keywords

  • Banach space;
  • measure of weak non-compactness;
  • James's Compactness Theorem

2010 Mathematics subject classification

  • Primary 46B50