The Closed Linear Span of { xk - ck } ; 30

Let { ck} ;” be a given real sequence. We wish to determine, in the first instance, easily verified conditions on {ck > F which imply that the sequence of functions { Xk ck > ;” IS total in C[O, 11; that is, that the closed linear span F{xk ck) is all of C[O, l] or, in other words, that every real function, continuous in [0, 11, is the limit, in the uniform norm, of a sequence of finite linear combinations of the xk ck. When this happens we refer to {ck};c as an approximating sequence. Since the sequence (.x”jF is total in C[O, 11, our problem is equivalent to demanding that the function f(x) E 1 belong to P{.x“ ck}. In this case we wish, in the second instance, to find an effective approximation to f(x) = 1 in the uniform norm on [O, l] by finite linear combinations of the 2~~ ck. An equivalent formulation of our problem, which is sometimes useful, is afforded by the following proposition. This proposition is an elementary application of the Hahn-Banach theorem.