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Remarks on remotal sets in vector valued function spaces. (English) Zbl 1207.46018

A subset \(E\) of a Banach space \(X\) is called remotal if for any \(x\in X\) there is a remotal point in \(E\). The present paper is concerned with remotal sets in the space \(L^p([0,1],X)\) of \(p\)-Bochner integrable functions (classes). In particular, it is proved that, if \( E\) is a finite set in a Banach space \(X\), then \(L^1([0,1],E)\) is remotal in \(L^1([0,1],X)\); if \(E\) is a closed bounded set in \(X\) such that the span of \(E\) is a finite-dimensional subspace of \(X\), then \(L^1([0,1],E)\) is remotal in \(L^1([0,1],X)\); also, if \(E\) is a closed bounded subset of a Banach space \(X\), then \(L^1([0,1],E)\) is remotal in \(L^1([0,1],X)\) if and only if \(L^p([0,1],E)\) is remotal in \(L^p([0,1],X)\), \(1 < p < \infty\).

MSC:

46B20 Geometry and structure of normed linear spaces
41A50 Best approximation, Chebyshev systems
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
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