Jm. Borwein et Jd. Vanderwerff, EPIGRAPHICAL AND UNIFORM-CONVERGENCE OF CONVEX-FUNCTIONS, Transactions of the American Mathematical Society, 348(4), 1996, pp. 1617-1631
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).