Another note on polynomial vs rational approximation

Let E be a subspace of C(X) and let X(E) = g/h:g. h is an element of E; h > 0}. We make a simple, yet intriguing observation: if zero is a best approximation to f from E, then zero is a best approximation to f from R(E). We also prove that if {E(n)} is dense in C(X) then for almost all f (in the sense of category).