The Basic Theorem and its Consequences

Abstract

Let T be a compact Hausdor 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 2 C(T ) from M to the case of finite T , that is to the case of approximation in l 1 (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 C0(T ) there exists a separating measure for M and f 2 C0(T )\M, the cardinality of whose support is not greater than dimM+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 2 C(T ). We also use the Basic Theorem to establish the suciency of Haar's condition for a subspace M of C(T ) to be Chebyshev.