On the Weierstrass-Stone Theorem Original Research Article

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 ƒ on S with the uniform norm ||ƒ|| = sup{||ƒ(t)||; t ∈ 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.