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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the behavior of weak convergence under nonlinearities and applications


Authors: Diego R. Moreira and Eduardo V. Teixeira
Journal: Proc. Amer. Math. Soc. 133 (2005), 1647-1656
MSC (2000): Primary 46B03, 46B10, 46B20
Published electronically: December 21, 2004
MathSciNet review: 2120260
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Abstract: This paper provides a sufficient condition to guarantee the stability of weak limits under nonlinear operators acting on vector-valued Lebesgue spaces. This nonlinear framework places the weak convergence in perspective. Such an approach allows short and insightful proofs of important results in Functional Analysis such as: weak convergence in $L^\infty$ implies strong convergence in $L^p$ for all $1\le p < \infty$, weak convergence in $L^1$ vs. strong convergence in $L^1$ and the Brezis-Lieb theorem. The final goal is to use this framework as a strategy to grapple with a nonlinear weak spectral problem on $W^{1,p}$.


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Additional Information

Diego R. Moreira
Affiliation: Department of Mathematics, University of Texas at Austin, RLM 12.128, Austin, Texas 78712-1082
Email: dmoreira@math.utexas.edu

Eduardo V. Teixeira
Affiliation: Department of Mathematics, University of Texas at Austin, RLM 9.136, Austin, Texas 78712-1082
Email: teixeira@math.utexas.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07876-1
PII: S 0002-9939(04)07876-1
Received by editor(s): April 24, 2003
Published electronically: December 21, 2004
Additional Notes: The second author is grateful for the financial support by CNPq - Brazil
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2004 American Mathematical Society