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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

   

 

Continuity of extremal elements in uniformly convex spaces


Author: Timothy Ferguson
Journal: Proc. Amer. Math. Soc. 137 (2009), 2645-2653
MSC (2000): Primary 30H05; Secondary 46B99
Published electronically: March 17, 2009
MathSciNet review: 2497477
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Abstract: In this paper, we study the problem of finding the extremal element for a linear functional over a uniformly convex Banach space. We show that a unique extremal element exists and depends continuously on the linear functional, and vice versa. Using this, we simplify and clarify Ryabykh's proof that for any linear functional on a uniformly convex Bergman space with kernel in a certain Hardy space, the extremal function belongs to the corresponding Hardy space.


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Additional Information

Timothy Ferguson
Affiliation: Department of Mathematics, University of Michigan, 2074 East Hall, 530 Church Street, Ann Arbor, Michigan 48109-1043
Email: tjferg@umich.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-09-09892-X
PII: S 0002-9939(09)09892-X
Received by editor(s): September 9, 2008
Published electronically: March 17, 2009
Communicated by: Mario Bonk
Article copyright: © Copyright 2009 Timothy Ferguson