Series of convex functions: subdifferential, conjugate and applications to entropy minimization

A formula for the sub-differential of the sum of a series of convex functions defined on a Banach space was provided by X. Y. Zheng in 1998. In this paper, besides a slight extension to locally convex spaces of Zhengs results, we provide a formula for the conjugate of a countable sum of convex functions. Then we use these results for calculating the sub-differentials and the conjugates in two situations related to entropy minimization, and we study a concrete example met in Statistical Physics.