(1) If f(x,y) is a real function and continuous on the closed rectangle X=[a,b]x[c,d] in the Euclidean plane R^n.
Show that f can be uniformly approximated on X by polynomials in x ad in y with real coefficients.
(2) Let X be the closed unit disc in the complex plane (denoted C).
Show that any function in C(X,C) can be uniformly approx on X by polynomials in z and conjugate z with complex coefficients.
(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).