On a theorem of Rudin

Abstract

We give short proofs of a theorem of Rudin about polynomial approximation in R 2 + n {R^{2 + n}} and a corollary of this theorem which says that any function algebra on [0, 1] generated by one complex-valued function and n real functions is all continuous functions. At the same time our proof shows that both results hold with n replaced by an arbitrary index set Λ \Lambda .</p