Posted by student on October 28, 2007, 6:33 pm I am getting confused over the hypotheses we need for the Stone-Weierstrass Theorem. In Royden, it seem that the only place he uses the inclusion of the constant functions is to prove that cl(A) is a lattice. But this is always true (he uses it only to show that |x| can be approximated by a polynmial, but by the classical Weierstrass theorem, there exists a sequence of polynomials P_n --> |x| uniformly, so P_n(0) --> 0, so P_n - P_n(0) --> 0 uniformly and has no constant term). Where else do we need the inclusion of the constant functions?
Hi Professor, 129
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