Approximation properties for spaces of Bochner integrable functions (Q472358)

Let \(X\) be a real Banach space with dual \(X^*\), \((\Omega,\mathcal A,\mu)\) a finite positive measure space and \(L^1(\mathcal A,X)\) the Banach space of all Bochner integrable functions defined on \(\Omega\) and taking values in \(X\). The author gives a simple proof of the following result: If the dual space \(X^*\) has the Radon-Nikodým property, then for every sub-\(\sigma\)-algebra \(\mathcal B\) of \(\mathcal A\) the space \(L^1(\mathcal B,X^*)\) is proximinal in \(L^1(\mathcal A,X^*)\). He also proves that if \(Y\) is a closed subspace of \(X\) such that \(L^1(\mathcal A,Y)\) is proximinal in \(L^1(\mathcal A,X)\), then for every sub-\(\sigma\)-algebra \(\mathcal B\) of \(\mathcal A\) the space \(L^1(\mathcal B,Y)\) is proximinal in \(L^1(\mathcal A,X).\) (A subset \(Z\) of a normed space \(X\) is called proximinal if every \(x\in X\) has a nearest point in \(Z\)). Another results concerns \(L\)-embedded spaces in the sense of \textit{P. Harmand} et al. [\(M\)-ideals in Banach spaces and Banach algebras. Berlin: Springer-Verlag (1993; Zbl 0789.46011)]. Let \(X\) be a Banach space such that the space \(L^1(\mathcal A,X)\) is \(L\)-embedded and let \(\mathcal A_n\) be a sequence of sub-\(\sigma\)-algebras increasing to the \(\sigma\)-algebra \(\mathcal A_\infty\subset \mathcal A\). If for \(f\in L^1(\mathcal A,X)\), \(f_n\in L^1(\mathcal A_n,X)\) is such that \(\|f-f_n\|=d(f,L^1(\mathcal A_n,X))\), \(n\in\mathbb N\), then the sequence \((f_n)\) is relatively weakly compact and any weak limit point of it is a best approximation for \(f\) in \(L^1(\mathcal A_\infty,X)\).