EPIGRAPHICAL AND UNIFORM-CONVERGENCE OF CONVEX-FUNCTIONS

We examine when a sequence of Isc convex functions on a Banach space c onverges uniformly on bounded sets (resp. compact sets) provided it co nverges Attouch-Wets (resp. Painleve-Kuratowski). We also obtain relat ed results for pointwise convergence and uniform convergence on weakly compact sets. Some known results concerning the convergence of sequen ces of linear functionals are shown to also hold for lsc convex functi ons. For example, a sequence of lsc convex functions converges uniform ly on bounded sets to a continuous affine function provided that the c onvergence is uniform on weakly compact sets and the space does not co ntain an isomorphic copy of l(1).