Epigraphical and Uniform Convergence of Convex Functions

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...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 1996-04, Vol.348 (4), p.1617-1631
Main Authors: Borwein, Jonathan M., Vanderwerff, Jon D.
Format: Article
Language:English
Subjects:
Banach space
Convexity
Frechet topologies
Indicator functions
Mathematical functions
Mathematical theorems
Perceptron convergence procedure
Topological theorems
Unit ball
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Summary: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}.
ISSN:0002-9947
1088-6850
DOI:10.1090/S0002-9947-96-01581-4