A quantitative version of James's compactness theorem (Q2891990)

In the last decade several authors -- see the comprehensive list in the references of the present reviewed item -- contributed to what can be described as ``quantification'' of several weak-compactness and related results -- essentially, describing how far the situation is from the accomplishment of the sought property (for example, ``measuring'' how much the \(w^*\)-closure in \(X^{**}\) of a bounded subset of a Banach space \(X\) ``leaves'' \(X\) tells how far from being \(w\)-compact the set is). Of course, those results, for the extreme case, include the ``classical'' theorem -- if the closure does not leave \(X\) at all the set is relatively \(w\)-compact. Naturally, this approach requires, in each case, the definition of the adequate ``non-compactness index''. One of them, arising from Grothendieck's double limit condition, is NEWLINE\[NEWLINE\gamma(H):=\{|\lim_{n}\lim_{m}x^*_m(x_n)-\lim_{m}\lim_{n}x^*_m(x_n)|:\;(x^*_m)\subset B_{X^*},\;(x_n)\subset H\}NEWLINE\]NEWLINE assuming the limits involved exist, where \(H\) is a bounded subset of a Banach space \(X\).NEWLINENEWLINEIn the present paper, the authors tackle the famous James compactness theorem by introducing the following new index associated to a given bounded subset \(H\) of a Banach space \(X\) NEWLINE\[NEWLINE\begin{multlined} \text{Ja}_X(H):=\inf\{\varepsilon>0:\;\text{for every }x^*\in X^*\\ \text{there is }x^{**}\in\overline{H}^{w^*}\;\text{such that }x^{**}(x^*)=\sup x^*(H)\;\text{and }d(x^{**},X)\leq\varepsilon\}. \end{multlined}NEWLINE\]NEWLINE The main result -- that includes James's compactness theorem -- reads: (Theorem 3.1) Let \(X\) be a Banach space and \(H\subset X\) be a bounded subset. Then \((1/2)\gamma(H)\leq \text{Ja}_{X}(H)\). The proof uses, among many other ingredients, results by Pryce in his study of James's theorem. It has the following remarkable consequence (here \(\hat d\) is the Hausdorff non-symmetrized distance): (Theorem 1.1) Let \(X\) be a Banach space, and let \(C\) be a closed convex bounded subset that is not \(w\)-compact. Let \(0\leq c<(1/2)\hat d(\overline{C}^{w^*},C)\) be arbitrary. Then there exists some \(x^*\in X^*\) such that for any \(x^{**}\in\overline{C}^{w^*}\) satisfying \(x^{**}(x^*)=\sup x^*(C)\) we have \(\text{dist}(x^{**},X)>c\). The authors pursue, additionally, a thorough comparison of all indices previously used in this direction (Corollary 3.5). Examples to ensure that estimates used are sharp are included, and several interesting open problems are scattered throughout the paper.