Gamma-Convergence of Convex Functions, Conjugates, and Subdifferentials

Rafael Correa; Pedro Pérez-Aros; José Pablo Santander

Paper

Gamma-Convergence of Convex Functions, Conjugates, and Subdifferentials

Abstract

This work establishes dual and subdifferential characterizations of

-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

when

is a

-finite measure and

spaces when

is an Eberlein compact topological space. It is shown that such a sequence

-converges in the strong topology to a limit function if and only if the sequence of Fenchel conjugates

-converges in the

-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--

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)].