A quantitative version of James's Compactness Theorem

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. 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 C w * whose distance to C is greater than r, then there is x * ∈ E * such that each x * * ∈ C w * 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.

Measures of weak non-compactness in spaces of nuclear operators

“…Another approach was used to prove a quantitative version of the Krein theorem -it was done independently in three papers [11,13,8] using different methods. The second approach inspired a fruitful research, the applications include quantitative versions of several classical theorems on weak compactness (Eberlein-Šmulyan theorem [2], Gantmacher theorem [3], James compactness theorem [7,14]), a characterization of subspaces of weakly compactly generated spaces [12] or a quantitative view on several properties of Banach spaces (Dunford-Pettis property [17], reciprocal Dunford-Pettis property [19], Banach-Saks property [5] etc. ).…”

An extension of James's compactness theorem

“…We have thus inductively constructed a sequence of positive scalars (b n ) satisfying (1) and (2). It follows that n b n is a convergent series and hence (3) is an immediate consequence of (2).…”

“…Subsequently, James [10] characterized the weakly compact subsets of a Banach space as the weakly closed, bounded subsets on which every bounded linear functional attains its supremum. James's proofs were technically quite demanding and there has been a considerable effort made for discovering a simpler proof ( [20], [3], [22], [6], [7], [5], [16], [12], [18], [19], [15], [2]). …”

“…The proof of James's theorem given in [16] was the first one to combine the results from [24], [23] and [8]. We finally mention paper [2] where the arguments of [20] are extended to give quantitative versions of James's theorem.…”

Measures of weak non-compactness in preduals of von Neumann algebras and JBW⁎-triples

“…They are used to prove more precise versions of known results and to establish new results as well. As an illustration we mention a fixed-point theorem [21], quantitative versions of Krein's theorem [26,31], James' compactness theorem [18,32], Eberlein-Šmulyan theorem [3] and Gantmacher's theorem [4].…”