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Various notions of best approximation property in spaces of Bochner integrable functions. (English) Zbl 1378.46029

Let \(C\) be a closed convex subset of a normed space \(X\) and \(P_C\) the metric projection on \(C\), \(P_C(x)=\{y\in C : \|x-y\|=d(x,C)\}\), and \(P_C(x,\delta)=\{y\in C : \|x-y\|\leq \delta +d(x,C)\}\), where \(\delta>0\). The set \(C\) is called proximinal if \(P_C(x)\neq\emptyset\) for all \(x\in X\) and strongly proximinal if it is proximinal and for every \(x\in X\) and \(\varepsilon >0\) there exists \(\delta>0\) such that \(P_C(x,\delta)\subseteq P_C(x)+\varepsilon B_X\), where \(B_X\) is the closed unit ball of \(X\). If \(Y\) is a closed subspace of \(X\), then \(Y\) is called (strongly) ball proximinal in \(X\) if its closed unit ball \(B_Y\) is (strongly) proximinal in \(X\). If \(Y\) is (strongly) ball proximinal in \(X\), then \(Y\) is (strongly) proximinal in \(X\), but the converses are not true.
The paper is concerned with ball proximinality, strong proximinality and strong ball proximinality in Lebesgue-Bochner spaces, namely, for the subspace \(L_p(I,Y)\) of \(L_p(I,X)\), where \(Y\) is a closed subspace of \(X\) and \(I\) is the interval \([0,1]\) with the Lebesgue measure. For instance, if \(Y\) is a separable proximinal subspace of \(X\), then:
(1) for every \(1\leq p\leq\infty,\, L_p(I,Y)\) is ball proximinal in \( L_p(I,X)\) iff \(Y\) is ball proximinal in \(X\);
(2) for every \(1\leq p<\infty,\, L_p(I,Y)\) is strongly proximinal (strongly ball proximinal) in \( L_p(I,X)\) iff \(Y\) is strongly proximinal (strongly ball proximinal) in \(X\).

MSC:

46E40 Spaces of vector- and operator-valued functions
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
46B20 Geometry and structure of normed linear spaces
46E15 Banach spaces of continuous, differentiable or analytic functions
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