Diego R. Moreira, Eduardo V. Teixeira
On the behavior of weak convergence under nonlinearities and applications
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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 puts 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 =< p < /infty, weak convergence in L^1 v.s. strong convergence in L^1 and 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.