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Extreme points in spaces of continuous functions. (English) Zbl 0832.46030
Summary: We study the \(\lambda\)-property for the space \({\mathcal C}(T, X)\) of continuous and bounded functions from a topological space \(T\) into a strictly convex Banach space \(X\). We prove that the \(\lambda\)-property for \({\mathcal C}(T, X)\) 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 \({\mathcal C}(T, X)\) 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.

MSC:
46E40 Spaces of vector- and operator-valued functions
46B20 Geometry and structure of normed linear spaces
46E15 Banach spaces of continuous, differentiable or analytic functions
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