There are several analogues of
the Weierstrass approximation theorem that characterize the uniform closure of a
C[X] submodule ℳ of the space C[X : E] of bounded continuous E-valued functions
on a compact space X. In this paper, a strong form of such a theorem is obtained
which is then applied to yield a characterization of all the functionals ϕ in the
dual of C[X : E] that are extreme among those of unit norm that vanish
on an arbitrary chosen ℳ. Each is determined by a point x0∈ X and a
unit functional L that is extreme in the annihilator of a closed subspace
M ⊂ E.
