The applications of the duality theorems to the approximation of functions

This study is concerned with the possibility of applying duality theorems of functional analysis to the approximation of functions. The duality theorems are first proved, using the Hahn-Banach theorem, and they are then reformulated in terms of quotient spaces. The general theory of the best approximation is then considered. For a subspace of M of X, we define SM as the intersection of M1 and S(the unit ball in X*). The duality theorems are used to characterise best approximations from M in terms of functionals from SM. Necessary and sufficient conditions in terms of SM for uniqueness of the best approximation are obtained. Since best approximation do not always exist, we then consider the more general case of good (specifically £-good) approximation. It has been shown that the maximum value of |X*(X)| over all functionals X* in SM exists and is equal to the least upper bound of |X(X)| over the set of extreme functionals of SM. The dual theorems are used to characterise £- good approximations from M in terms of functionals from the set of extreme functionals of SM. We then apply these results to the Chebyshev problem, reformulating and proving the Chebyshev alternation theorem with the help of the duality theorems. The alternation of best approximation in the plane is then considered. It is shown that in a particular case the error function of the best approximation can be considered to alternate at the four vertices of rectangle in the plane, although the best approximation in this case is not unique.