📝 SummarXiv

Abstract

This work establishes dual and subdifferential characterizations of Unicode ({\Gamma})-convergence for sequences of proper convex lower semicontinuous functions in weakly compactly generated Banach spaces. It is shown that such a sequence Unicode ({\Gamma})-converges in the strong topology to a limit function if and only if the sequence of conjugates Unicode ({\Gamma})-converges in the $w^*$-topology to the Fenchel conjugate. 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. The analysis develops new arguments in separable Banach spaces and extends them to the general framework through separable reduction techniques. Additionally, we introduce several new rich families of convex functions that exhibit various separable reduction properties.