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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Epigraphical and Uniform Convergence of Convex Functions


Authors: Jonathan M. Borwein and Jon D. Vanderwerff
Journal: Trans. Amer. Math. Soc. 348 (1996), 1617-1631
MSC (1991): Primary 46A55, 46B20, 52A41
MathSciNet review: 1344203
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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. 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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Additional Information

Jonathan M. Borwein
Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, BC, Canada V5A 1S6
Email: jborwein@cecm.sfu.ca

Jon D. Vanderwerff
Affiliation: Department of Mathematics, Walla Walla College, College Place, Washington 99324
Email: vandjo@wwc.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01581-4
PII: S 0002-9947(96)01581-4
Keywords: Epi-convergence, lsc convex function, uniform convergence, pointwise convergence, Attouch-Wets convergence, Painlev\'{e}-Kuratowski convergence, Mosco convergence
Received by editor(s): January 17, 1995
Received by editor(s) in revised form: April 3, 1995
Additional Notes: The first author’s research supported in part by an NSERC research grant and by the Shrum endowment
Article copyright: © Copyright 1996 American Mathematical Society