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Complex extremal structure in spaces of continuous functions. (English) Zbl 0920.46028
Let $$T$$ be a completely regular space, $$X$$ be a complex normed space and $$C(T,X)$$ be the space of all continuous and bounded $$X$$-valued functions on $$T$$ equipped with the uniform norm. It is shown that if the dimension of $$X$$ is finite then every function in the unit ball of $$C(T,X)$$ is expressible as a convex combination of three unitary functions if and only if $$\dim T< \dim X_R$$ (here $$\dim T$$ denotes the covering dimension of $$T$$ and $$X_R$$ denotes $$X$$ considered as a real normed space). If $$X$$ is an infinite-dimensional space the same kind decomposition is possible without restrictions for $$T$$. Moreover, it is proved that the identity mapping on the unit ball of an infinite-dimensional complex normed space $$X$$ can be expressed as the average of three retractions of the unit ball of $$X$$ onto the unit sphere of $$X$$. The case, when $$X$$ is a complex strictly convex normed space, is considered separately.
Reviewer: Mati Abel (Tartu)

##### MSC:
 4.6e+41 Spaces of vector- and operator-valued functions 4.6e+16 Banach spaces of continuous, differentiable or analytic functions
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