An elementary proof of James’ characterisation of weak compactness. II. (English) Zbl 1372.46006
James’ characterisation of weak compactness states that a closed bounded convex subset of a real Banach space is weakly compact if it is a boundary for its weak\(^*\) closure in the bidual. In this context, V. P. Fonf and J. Lindenstrauss [Isr. J. Math. 136, 157–172 (2003; Zbl 1046.46014)] invented the notion of (I)-generation and showed that a boundary is (I)-generating. (For the precise definitions, see the paper.)
The authors follow the line of reasoning of Part I [Bull. Aust. Math. Soc. 84, No. 1, 98–102 (2011; Zbl 1227.46014)] but with the difference that in the main part the Fonf-Lindenstrauss result is reproved in a more concise, more elementary way, without resorting to theorems like the ones of Krein-Milman and Bishop-Phelps.
As in Part I, there is then a short elementary way from (I)-generation to James’ theorem for those Banach spaces whose dual unit ball is weak\(^*\) sequentially compact. (The authors mention that in this way their proof recovers James’ theorem for a great class of Banach spaces but, for example, not for \(l^{\infty}\)).
The authors follow the line of reasoning of Part I [Bull. Aust. Math. Soc. 84, No. 1, 98–102 (2011; Zbl 1227.46014)] but with the difference that in the main part the Fonf-Lindenstrauss result is reproved in a more concise, more elementary way, without resorting to theorems like the ones of Krein-Milman and Bishop-Phelps.
As in Part I, there is then a short elementary way from (I)-generation to James’ theorem for those Banach spaces whose dual unit ball is weak\(^*\) sequentially compact. (The authors mention that in this way their proof recovers James’ theorem for a great class of Banach spaces but, for example, not for \(l^{\infty}\)).
Reviewer: Hermann Pfitzner (Orléans)
MSC:
| 46A50 | Compactness in topological linear spaces; angelic spaces, etc. |
| 46B22 | Radon-Nikodým, Kreĭn-Milman and related properties |
| 46B10 | Duality and reflexivity in normed linear and Banach spaces |
Keywords:
weakly compact sets; James’ theorem; boundaries; (I)-generation; weak\(^*\) sequentially compact dual unit ballReferences:
| [1] | R.Deville, G.Godefroy and V.Zizler, Smoothness and Renormings in Banach Spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, 64 (Longman Scientific and Technical, Harlow, 1993), (copublished in the United States with John Wiley, New York). · Zbl 0782.46019 |
| [2] | J.Hagler and F.Sullivan, ‘Smoothness and weak* sequential compactness’, Proc. Amer. Math. Soc.78 (1980), 497-503. · Zbl 0463.46010 |
| [3] | R. C.James, ‘Weakly compact sets’, Trans. Amer. Math. Soc.113 (1964), 129-140. · Zbl 0129.07901 |
| [4] | D.Larman and R. R.Phelps, ‘Gateaux differentiability of convex functions on Banach spaces’, J. Lond. Math. Soc. (2)20 (1979), 115-127.10.1112/jlms/s2-20.1.115 · Zbl 0431.46033 |
| [5] | W. B.Moors, ‘An elementary proof of James’ characterisation of weak compactness’, Bull. Aust. Math. Soc.84 (2011), 98-102. · Zbl 1227.46014 |
| [6] | J. D.Pryce, ‘Weak compactness in locally convex spaces’, Proc. Amer. Math. Soc.17 (1966), 148-155. · Zbl 0141.11702 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
