Epigraphical and Uniform Convergence of Convex Functions

Borwein, Jonathan; Vanderwerff, Jon

doi:10.1090/S0002-9947-96-01581-4

Epigraphical and Uniform Convergence of Convex Functions
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by Jonathan M. Borwein and Jon D. Vanderwerff PDF

Trans. Amer. Math. Soc. 348 (1996), 1617-1631 Request permission

Abstract:

We examine when a sequence of lsc convex functions on a Banach space converges uniformly on bounded sets (resp. compact sets) provided it converges Attouch-Wets (resp. Painlevé-Kuratowski). We also obtain related results for pointwise convergence and uniform convergence on weakly compact sets. Some known results concerning the convergence of sequences of linear functionals are shown to also hold for lsc convex functions. For example, a sequence of lsc convex functions converges uniformly on bounded sets to a continuous affine function provided that the convergence is uniform on weakly compact sets and the space does not contain an isomorphic copy of $\ell _{1}$.

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