Linear combinations of partitions of unity with restricted supports

Abstract

Abstract. Given a locally finite open covering C of a normal space X and a Hausdorff topological vector space E, we characterize all continuous functions f : X → E which admit a representation f = C∈C a C ϕ C with a C ∈ E and a partition of unity {ϕ C : C ∈ C} subordinate to C.As an application, we determine the class of all functions f ∈ C(|P|) on the underlying space |P| of a Euclidean complex P such that, for each polytope P ∈ P, the restriction f | P attains its extrema at vertices of P . Finally, a class of extremal functions on the metric space (is characterized, which appears in approximation by so-called controllable partitions of unity. [10], p. 352). For a fixed locally finite open covering C of X and for a Hausdorff topological vector space E, we want to characterize all continuous functions f : X → E which admit a representation f = C∈C a C ϕ C with vectors a C ∈ E and a partition of unity {ϕ C : C ∈ C} subordinate to C. The following theorem solves this question for normal spaces X. We shall see that the claim of this very general theorem can even be used for characterizing normal spaces among T 1 -spaces. Besides that, we shall present an application concerning continuous real-valued functions on polytopes and on Euclidean complexes in R m as they appear in optimization theory (see Section 2).