2025/09/26 by Correa, Rafael, Pérez-Aros, Pedro, Santander, José Pablo
#FOS: Mathematics #Functional Analysis (math.FA) #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.2509.21863
This work establishes dual and subdifferential characterizations of Γ-convergence for sequences of proper convex lower semicontinuous functions in weakly compactly generated Banach spaces, which include separable spaces and also reflexive ones, as well as L1(μ) when μ is a σ-finite measure and C(K) spaces when K is an Eberlein compact topological space. It is shown that such a sequence Γ-converges in the strong topology to a limit function if and only if the sequence of Fenchel conjugates Γ-converges in the w^∗-topology to the conjugate of the limit function. It is further proved that both conditions are equivalent to the graphical convergence of the associated subdifferentials with respect to the strong--w^∗ product topology. Counterexamples demonstrate that these equivalences break down outside the weakly compactly generated setting. Furthermore, our approach develops a rich family in weakly compactly generated spaces similar to the one recently used to characterize Asplund spaces [C'uth and Fabian (2016)].