On the behavior of weak convergence under nonlinearities and applications

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-infinity implies strong convergence in L-p for all 1 <= p < infinity, 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,W-p.