Topological spaces which admit unisolvent systems

Abstract

Introduction. We begin with a definition. Let X be a topological space and let e(X) be the set of all continuous functions which map X into R, the real numbers. Let J = {f1, ,f...4J be a set of n distinct functions in W(X) such that each nonzero linear combination of these functions over R has at most n 1 zeros in X. It is clear that this property of F is equivalent to the property that given any set of n distinct points {x1,-.,xn} in X, the matrix ||fi(xj) || has an inverse. We may call such a set b5 a real-valued unisolvent system of order n on X, or for brevity, an n-R-U system on X(1). For examples, confer (1.6) and (2.5) below. The purpose of this paper is to characterize topologically those spaces which admit n-R-U systems. It should be noted that any nonempty topological space of less than n points has an n-R-U system and that any nonempty topological space whatsover has a 1-R-U system. Consequently in this paper we shall consider only spaces with at least n points, where n > 2. A summary of this problem's background is useful. For this the following definition is needed. Let X4 be an n-dimensional vector R-subspace of ro(X), where X is a locally compact Hausdorff space and TO(X) is the vector R-subspace of W(X) containing all functions which vanish at infinity (i.e., all functions f in W(X) such that for each ? > 0, the set {x E X: If(x) j >, } is compact). X' is said to be a Haar subspace of TO(X) if given anyf in TO(X) there is a unique gf in X such that