Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Extreme points in spaces of continuous functions

Authors: V. I. Bogachev, J. F. Mena-Jurado and J. C. Navarro Pascual
Journal: Proc. Amer. Math. Soc. 123 (1995), 1061-1067
MSC: Primary 46E40; Secondary 46B20, 46E15, 54C20
MathSciNet review: 1204371
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Abstract: We study the $ \lambda $-property for the space $ \mathfrak{C}(T,X)$ of continuous and bounded functions from a topological space T into a strictly convex Banach space X. We prove that the $ \lambda $-property for $ \mathfrak{C}(T,X)$ is equivalent to an extension property for continuous functions of the pair (T, X). We show also that, when X has even dimension, the $ \lambda $-property is equivalent to the fact that the unit ball of $ \mathfrak{C}(T,X)$ is the convex hull of its extreme points and that this last property is true if X is infinite dimensional. As a result we get that the identity mapping on the unit ball of an infinite-dimensional strictly convex Banach space can be expressed as the average of four retractions of the unit ball onto the unit sphere.

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Keywords: $ \lambda $-property, uniform $ \lambda $-property, extreme point, strictly convex space, covering dimension
Article copyright: © Copyright 1995 American Mathematical Society