Approximation of vector-valued continuous functions

Abstract

The results of this article are important for proving Riesz-type representation theorems for spaces of continuous func- tions with values in a topological vector space. It is well known that every continuous function with compact support from a locally compact Hausdorff space to a locally convex space can be uniformly approximated by continuous functions with finite-dimensional range. We give several conditions sufficient for this to be true with- out convexity. This problem is related to a vector-valued Tietze extension problem, and we give a new proof of a theorem of Dugundji, Arens, and Michael in this area, using topological tensor products. Let Pbe a compact Hausdorff space, E a topological vector space (TVS) over either the real or complex field, and C(T, E) the space of continuous functions from T to E, with the topology of uniform convergence. When E is the scalar field we write C(T) instead of C(T, E). For each a e C(T) and x e .£7 the function / -*■ a(t)x from 7 to E, denoted by a ® x, is contin- uous. The linear span of these functions in C(T, E) is the set of all finite sums 2 ai ® xi with e C(T) and x, e E and is isomorphic to the alge- braic tensor product C(T) ® E. If E is locally convex, then C(T) ® E is dense in C(T, E) (4, Chapter III, §1, Proposition 1 and Lemma 2). This property, the density property, is of prime importance in the representation of linear functionals and operators on C(T, E) by vector measures (12). The sufficient conditions given in this article have been used by the author (11) to extend these representations to the case when E is not assumed to be locally convex. It is not known if any space C(T, E) fails to have the density property.