Approximation by Extreme Functions

Abstract

ForTa topological space andXa real normed space,Y=C(T,X) denotes the space of continuous and bounded functions fromTintoXendowed with the sup norm. We calculate a formula for the distanceα(f) fromfinYto the setY−1of functions inYwhich have no zeros. Namely, we prove thatα(f) is the infimum of numbersδ>0 for which the continuous functiont↦f(t)/‖f(t)‖ defined for everytwith ‖f(t)‖⩾δhas a continuous extensionefromTinto the unit sphere ofX. This permits us to get the general expression of the Aron–Lohmanλ-function ofYwhenXis strictly convex. We show that any function inYhas a best approximation inY−1which can be chosen to have the least possible norm. IfXis strictly convex andE(Y) denotes the set of extreme points of the unit ball ofY, this fact enables us to prove that dist(f,E(Y))=max{1−m(f)+α(f), ‖f‖−1} ∀f∈Y, wherem(f)=inf {‖f(t)‖:t∈T}. Moreover, we show thatE(Y) is proximinal inY−1and give sufficient conditions under whichfinY\\Y−1admits a best approximation inE(Y).