Clearly if then we can define a map from by sending an arbitrary ordered pair to the difference between the kth coordinate of it and the kth coordinate of , thus if is that function we have that . Thus, the set of all polynomials separates points and since clearly there are infinitely many constant functions in your space it follows from the S.W.T. that the closure of the set of all polynomials is . Conclude

Similar, let's see some work (you will clearly need to look at the complex version of the S.W.T)(2) Let X be the closed unit disc in the complex plane (denotedC).

Show that any function in C(X,C) can be uniformly approx on X by polynomials in z and conjugate z with complex coefficients.

Slightly trickier, ideas?(3) Let X and Y be compact Hausdorff spaces, and f a function in C(XxY,C).

Show that f can be uniformly approx. by functions of the form Sum(from 1 to n)(f_i)(g_i), where the f_i's are in C(X,C) and the g_i's are in C(Y,C).