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Extreme points and retractions in Banach spaces. (English) Zbl 0901.46015
Summary: For \(T\) a completely regular topological space and \(X\) a strictly convex Banach space, we study the extremal structure of the unit ball of the space \(C(T,X)\) of continuous and bounded functions from \(T\) into \(X\). We show that when \(\dim X\) is an even integer then every point in the unit ball of \(C(T,X)\) can be expressed as the average of three extreme points if, and only if, \(\dim T<\dim X\), where \(\dim T\) is the covering dimension of \(T\). We also prove that, if \(X\) is infinite-dimensional, the aforementioned representation of the points in the unit ball of \(C(T,X)\) is always possible without restrictions on the topological space \(T\). Finally, we deduce from the above result that the identity mapping on the unit ball of an infinite-dimensional strictly convex Banach space admits a representation as the mean of three retractions of the unit ball onto the unit sphere.

MSC:
46B20 Geometry and structure of normed linear spaces
46E40 Spaces of vector- and operator-valued functions
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