Applications The Stone–Weierstrass theorem can be used to prove the following two statements, which go beyond Weierstrass's result. If is a continuous real-valued function defined on the set and , then there exists a polynomial function in two variables such that for all in and in . If and are two compact Hausdorff spaces and is a continuous function, then for every there exist and continuous functions on and continuous functions on such that .
applications the stone–weierstrass theorem can be used to prove the following two statements which go beyond weierstrass's result if is a continuous real-valued function defined on the set and  then there exists a polynomial function in two variables such that for all in and in  if and are two compact hausdorff spaces and is a continuous function then for every there exist and continuous functions on and continuous functions on such that 