A Bornological Approach to Rotundity and Smoothness Applied to Approximation

In this paper we are interested in examining the geometry of a bounded convex function over a Banach space via its subdifferential mapping. We will consider two concepts. The first is the single valuedness and continuity of the subdifferential mapping, and the second is the single valuedness and the continuity of the “inverse” of this mapping. The smoothness of f is important for the first concept as the convexity of f is for the second. We generalize some of the well known results on upper semi-continuity of the subdifferential mapping, and we introduce a bornological approach to convexity, which allows us to draw very nice parallels for the continuity of the inverse mapping with the coresponding concept for the well understood subdifferential mapping. The theory developed allows us to give a local Smulyan result in which the convexity at a point on the unit sphere is characterized by the uniform smoothness of the subdifferential of this point, and to give the smoothness of the primal norm at a point in terms of the convexity of the dual norm about the subdifferential of that point. As the title implies we will place special emphasis on the approximation of this convex function by a sequence of such functions and derive conditions, which ensure satisfactory approximation of the subdifferential and “inverse” mappings.