Best simultaneous approximation in function and operator spaces. (English) Zbl 1248.41029

Summary: Let \(Z\) be a Banach space and \(G\) be a closed subspace of \(Z\). For \(f_1,f_2\in Z\), the distance from \(f_1\), \(f_2\) to \(G\) is defined by \(d(f_1,f_2,G)= \text{inf}_{f\in G}\max\{\| f_1- f\|,\,\|_2- f\|\}\). An element \(g^*\in G\) satisfying \[ \max\{\| f_1- g^*\|,\,\| f_2- g^*\|\}= \underset{f\in G}{}{\text{inf}}\, \max\{\| f_1- f\|,\, \| f_2- f\|\} \] is called a best simultaneous approximation for \(f_1\), \(f_2\) from \(G\).
In this paper, we study the problem of best simultananeous approximation in the space of all continuous \(X\)-valued functions on a compact Hausdorff space \(S\); \(C(S,X)\), and the space of all bounded linear operators from a Banach space \(X\) into a Banach space \(Y\); \(L(X,Y)\).


41A28 Simultaneous approximation