Stone-Weierstrass Theorem

If  is any compact space, let  be a subalgebra of the algebra  over the reals  with binary operations  and . Then, if  contains the constant functions and separates the points of  (i.e., for any two distinct points  and  of , there is some function  in  such that ),  is dense in  equipped with the uniform norm.

This theorem is a generalization of the Weierstrass approximation theorem.

REFERENCES:

Cullen, H. F. "The Stone-Weierstrass Theorem" and "The Complex Stone-Weierstrass Theorem." In Introduction to General Topology. Boston, MA: Heath, pp. 286-293, 1968.