Moreover, this p is unique, and is characterized by this property. In this paper we shall give a generalization of this result (except for uniqueness, which does not hold) on a locally compact space X. Let B = CiX) be the Banach space of all continuous real functions on X, vanishing at infinity, with ||/||=max {|/(x)| : xQX}. Let gi, • • • i gn be re fixed functions in B, linearly independent, and let P— V{gi, • • • , gn}The Chebychev approximation problem is to find p0 in E such that [|/-£o|| ^||/-/>|| for all p in E. We can then assert: