Best simultaneous approximation in \(L^{p}(I,E)\)
Let \(G\) be a subspace of a Banach space \(E\) and, for \(1\leq p< \infty\), let \(L^p([0,1],E)\) denote the space of all \(p\)-Bochner integrable functions on \([0,1]\) with values in \(E\). Let \(N(\cdot,\cdot)\) be a norm on \(\mathbb{R}^2\) such that \(N(x',z')\geq N(x,y)\) whenever \(x'\geq x\geq 0\) and \(y'\geq y\geq 0\). Then \(L^p([0,1],G)\) is said to be \(N\)-simultaneously proximinal in \(L^p([0, 1],E)\) if for each pair \(f_1,f_2\) in \(L^p([0,1],E)\) there exists \(g\in L^p([0, 1],G)\) such that, for all \(h\in L^p ([0,1],G)\), \[ N\bigl(\|f_1-g \|,|f_2-g\|\bigr)\leq N\bigl(\|f_1-h \|,\|f_2-h \|\bigr), \] that is, \(g\) is an \(N\)-simultaneous best approximation to the pair \(f_1,f_2\). The main result of the paper is that if \(G\) is reflexive and \(1\leq p<\infty\) then \(L^p([0,1],G)\) is \(N\)-simultaneously proximinal in \(L^p([0,1],E)\).
scientific article; zbMATH DE number 1800269