Let X be a Banach space and X' its continuous dual. C(X) (resp. C(X')) denotes the set of nonempty convex closed subsets of X (resp. ω*-closed subsets of X') endowed with the topology of uniform convergence of distance functions on bounded sets. This topology reduces to the Hausdorff metric topology on the closed and bounded convex sets [16] and in general has a Hausdorff-like presentation [11]. Moreover, this topology is well suited for estimations and constructive approximations [6-9]. We prove here, that under natural qualification conditions, the stability of the convergence associated to the topology defined on C(X) (resp. C(X')) is preserved by a class of linear transformations. Building on these results, and by identifing each convex function with its epigraph, the stability at the functional level is acquired towards some operations of convex analysis which play a basic role in convex optimization and duality theory. The key hypothesis in the qualification conditions ensuring the functional stability is the notion of inf-local compactness of a convex function introduced in [28] and expressed in the space X' by the quasi-continuity of its conjugate. Then we generalize the stability results of McLinden and Bergstrom [31] and the ones of Beer and Lucchetti [17] in infinite dimension case. Finally we give some applications in convex optimization and mathematical programming in general Banach spaces.