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 topologyof uniform convergence of distance functions on bounded sets. This topologyreduces to the Hausdorff metric topology on the closed and bounded convexsets [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 ofthe convergence associated to the topology defined on C(X) (resp. C(X')) is preserved by aclass of linear transformations. Building on these results, and byidentifing each convex function with its epigraph, the stability at thefunctional level is acquired towards some operations of convex analysiswhich play a basic role in convex optimization and duality theory. The keyhypothesis in the qualification conditions ensuring the functional stabilityis the notion of inf-local compactness of a convex function introduced in [28] and expressed in the space X' by the quasi-continuity of itsconjugate. Then we generalize the stability results of McLinden andBergstrom [31] and the ones of Beer and Lucchetti [17] in infinite dimensioncase. Finally we give some applications in convex optimization andmathematical programming in general Banach spaces.