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Remotal sets in vector valued function spaces. (English) Zbl 1118.46026

Let \(X\) be a Banach space. A bounded set \(E \subset X\) is said to be remotal if for every \(x \in X\) there exists a \(z \in E\) such that \(\| x-z\| = \sup \{\| x-y\| : y\in E\}\). This paper deals with remotal sets in the space of Bochner integrable functions \(L^1([0,1],\lambda,X)\) of the form \(L^1([0,1],\lambda,E)\) for a remotal set \(E \subset X\), where \(\lambda\) denotes the Lebesgue measure. Clearly, since remotal sets are bounded, one cannot expect such a result when the measure has infinitely many atoms (Theorem 2.2), unless \(E = \{0\}\). The authors first obtain a distance estimate, \(\sup \{\| f-g\| : g \in L^1([0,1],\lambda,E)\} =\int \sup\{\| f(s)-e\| : e \in E\}\,d\lambda(s)\) and use it to show that when \(E\) is a closed and bounded convex set such that its closed linear span is finite-dimensional then \(L^1([0,1],\lambda,E)\) is a remotal set.

MSC:

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