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 (C.T. 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

On the defect of compactness in Banach spaces / Solimini, Sergio Fausto; Tintarev, C.. - In: COMPTES RENDUS MATHEMATIQUES DE L'ACADEMIE DES SCIENCES. - ISSN 0706-1994. - 353:10(2015), pp. 899-903. [10.1016/j.crma.2015.07.011]

On the defect of compactness in Banach spaces

SOLIMINI, Sergio Fausto;
2015-01-01

Abstract

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 (C.T. 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
2015
On the defect of compactness in Banach spaces / Solimini, Sergio Fausto; Tintarev, C.. - In: COMPTES RENDUS MATHEMATIQUES DE L'ACADEMIE DES SCIENCES. - ISSN 0706-1994. - 353:10(2015), pp. 899-903. [10.1016/j.crma.2015.07.011]
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11589/7960
Citazioni
  • Scopus 3
  • ???jsp.display-item.citation.isi??? 3
social impact