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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
      Additional Information: pubdom FALSE
      Uncontrolled Keywords: epi-convergence, lsc convex function, uniform convergence, pointwise convergence, Attouch-Wets convergence, Painleve-Kuratowski convergence, Mosco convergence
      Subjects: 46-xx Functional analysis > 46Axx Topological linear spaces and related structures
      46-xx Functional analysis > 46Bxx Normed linear spaces and Banach spaces; Banach lattices
      52-xx Convex and discrete geometry > 52Axx General convexity
      Faculty: UNSPECIFIED
      Depositing User: Users 1 not found.
      Date Deposited: 17 Nov 2003
      Last Modified: 13 Sep 2014 21:18
      URI: https://docserver.carma.newcastle.edu.au/id/eprint/94

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