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On the defect of compactness in Banach spaces
Politecn Bari, I-70126 Bari, Italy..
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
2015 (English)In: Comptes rendus. Mathematique, ISSN 1631-073X, E-ISSN 1778-3569, Vol. 353, no 10, 899-903 p.Article in journal (Refereed) PublishedText
Abstract [en]

Concentration compactness methods improve convergence for bounded sequences in Banach spaces beyond the weak-star convergence provided by the Banach-Alaoglu theorem. A further improvement of convergence, known as profile decomposition, is possible up to defect of compactness, a series of "elementary concentrations" defined relative to the action of some group of linear isometric operators. This note presents a general profile decomposition for uniformly convex and uniformly smooth Banach spaces, generalizing the result of one of the authors (S.S.) for Sobolev spaces and of the other (CT. jointly with I. Schindler) for general Hilbert spaces. Unlike in the Hilbert space case, profile decomposition is based not on weak convergence, but on a different mode of convergence, called polar convergence, which coincides with weak convergence if and only if the norm satisfies the known Opial condition, used in the context of fixed point theory for nonexpansive maps.

Place, publisher, year, edition, pages
2015. Vol. 353, no 10, 899-903 p.
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URN: urn:nbn:se:uu:diva-272403DOI: 10.1016/j.crma.2015.07.011ISI: 000366081200007OAI: oai:DiVA.org:uu-272403DiVA: diva2:897356
Available from: 2016-01-25 Created: 2016-01-13 Last updated: 2016-04-13

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