On the existence of optimal affine methods for approximating linear functionals (Q1100370)

Let F be a set in a linear space and S, \(x_ 1,...,x_ n\) functionals on F, \(f\in F\). To construct an optimal method for approximation S(f) means to find a function \(\theta_ 0\) minimizing \(\sup_{f\in F}| S(f)-\theta (x_ 1(f),...,x_ n(f))|\) over the set of all real valued functions \(\theta\). A constructive proof of the existence of an affine function \(\theta_ 0(y_ 1,...,y_ n)=p_ 0+p_ 1y_ 1+...+p_ ny_ n\) which is optimal for approximation of a linear functional on a convex set is given.