# Epigraphical and Uniform Convergence of Convex Functions

Borwein, Jonathan M. and Vanderwerff, Jon D. (1996) Epigraphical and Uniform Convergence of Convex Functions. Transactions of the American Mathematical Society, 348 (4). pp. 1617-1631.

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## 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. Painlev\'e-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 $\ell_1$.

Item Type: Article pubdom FALSE epi-convergence, lsc convex function, uniform convergence, pointwise convergence, Attouch-Wets convergence, Painleve-Kuratowski convergence, Mosco convergence 46-xx Functional analysis > 46Axx Topological linear spaces and related structures46-xx Functional analysis > 46Bxx Normed linear spaces and Banach spaces; Banach lattices52-xx Convex and discrete geometry > 52Axx General convexity UNSPECIFIED Users 1 not found. 17 Nov 2003 13 Sep 2014 21:18 https://docserver.carma.newcastle.edu.au/id/eprint/94

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