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 $\ell _{1}$.References
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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
- 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
- © Copyright 1996 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 348 (1996), 1617-1631
- MSC (1991): Primary 46A55, 46B20, 52A41
- DOI: https://doi.org/10.1090/S0002-9947-96-01581-4
- MathSciNet review: 1344203