ON THE WEIERSTRASS-STONE THEOREM

Let S be a compact Hausdorff space, and let E be a normed space over the reals. Let C(S; E) be the linear space of all E-valued continuous functions f on S with the uniform norm \\f\\ = sup{\\ f(t)\\; t epsilon S}. When E = R, the Weierstrass-Stone Theorem describes the uniform closure of a subalgebra of C(s;R). We extend this classical result in two ways: we admit vector-valued functions and describe the uniform closure of arbitrary subsets of c(S;E). The classical Weierstrass-Stone Theorem is obtained as a corollary, without Zorn's Lemma. (C) 1994 Academic Press, Inc.