Approximation by Extreme Functions

✍ Scribed by A Jiménez-Vargas; J.F Mena-Jurado; J.C Navarro-Pascual

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 130 KB
• Volume: 97
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1996.3258

✦ Synopsis

For T a topological space and X a real normed space, Y=C(T, X) denotes the space of continuous and bounded functions from T into X endowed with the sup norm. We calculate a formula for the distance :( f ) from f in Y to the set Y &1 of functions in Y which have no zeros. Namely, we prove that :( f ) is the infimum of numbers $>0 for which the continuous function t [ f (t)Â& f (t)& defined for every t with & f (t)& $ has a continuous extension e from T into the unit sphere of X. This permits us to get the general expression of the Aron Lohman *-function of Y when X is strictly convex. We show that any function in Y has a best approximation in Y &1 which can be chosen to have the least possible norm. If X is strictly convex and E(Y) denotes the set of extreme points of the unit ball of Y, this fact enables us to prove that dist(

and give sufficient conditions under which f in Y"Y &1 admits a best approximation in E(Y).

1999 Academic Press

1. INTRODUCTION AND NOTATION

Throughout this paper the letter T stands for a topological space, while X denotes a real normed space. B(X), S(X), and E(X) are the closed unit