The Basic Theorem and its Consequences

Let T be a compact Hausdorff topological space and let M denote an n–dimensional subspace of the space C(T), the space of real–valued continuous functions on T and let the space be equipped with the uniform norm. Zukhovitskii [7] attributes the Basic Theorem to E.Ya.Remez and gives a proof by duality. He also gives a proof due to Shnirel’man, which uses Helly’s Theorem, now the paper obtains a new proof of the Basic Theorem. The significance of the Basic Theorem for us is that it reduces the characterization of a best approximation to f ε C(T) from M to the case of finite T, that is to the case of approximation in l^{ω}(r). If one solves the problem for the finite case of T then one can deduce the solution to the general case. An immediate consequence of the Basic Theorem is that for a finite dimensional subspace M of C_{0}(T) there exists a separating measure forMand f ε C_{0}(T)M, the cardinality of whose support is not greater than dim M+1. This result is a special case of a more general abstract result due to Singer [5]. Then the Basic Theorem is used to obtain a general characterization theorem of a best approximation from M to f ε C(T). We also use the Basic Theorem to establish the sufficiency of Haar’s condition for a subspace M of C(T) to be Chebyshev.