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

 

On the extremal structure of
the unit balls of Banach spaces of
weakly continuous functions and their duals


Authors: Zhibao Hu and Mark A. Smith
Journal: Trans. Amer. Math. Soc. 349 (1997), 1901-1918
MSC (1991): Primary 46B20, 46E40
MathSciNet review: 1407701
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Abstract: A sufficient and then a necessary condition are given for a function to be an extreme point of the unit ball of the Banach space $C(K,(X,w))$ of continuous functions, under the supremum norm, from a compact Hausdorff topological space $K$ into a Banach space $X$ equipped with its weak topology $w$. Strongly extreme points of the unit ball of $C(K,(X,w))$ are characterized as the norm-one functions that are uniformly strongly extreme point valued on a dense subset of $K$. It is shown that a variety of stronger types of extreme points (e.g. denting points) never exist in the unit ball of $C(K,(X,w))$. Lastly, some naturally arising and previously known extreme points of the unit ball of $C(K,(X,w))^{*}$ are shown to actually be strongly exposed points.


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Additional Information

Zhibao Hu
Affiliation: Division of Mathematics, El Paso Community College, Valle Verde Campus, P.O. Box 20500, El Paso, Texas 79998

Mark A. Smith
Affiliation: Department of Mathematics and Statistics, Miami University, Oxford, Ohio 45056

DOI: http://dx.doi.org/10.1090/S0002-9947-97-01903-X
PII: S 0002-9947(97)01903-X
Keywords: Extreme points, strongly extreme points, points of continuity, denting points, spaces of weakly continuous functions
Received by editor(s): November 9, 1995
Additional Notes: The second author was supported in part by a Miami University Summer Research Grant.
Article copyright: © Copyright 1997 American Mathematical Society