You need to be more formal. If (continous functions) were to be a finite dimensionsal vector space then every subspace would be finite dimensional also, with a smaller degree (this is a theorem). But then (polynomials functions) would be finite dimensional. But that is impossible. Why?
Again, why is P[0,1] infinite? You need to be more formal. Here is how you show it cannot have a finite basis. Suppose that P[0,1] has a finite basis where are polynomials. Then it would mean any polynomial in P[0,1] can be expressed in the form . Let be a polynomial with a larger degree then all of the polynomials. Then is impossible because polynomials can only be equal if their have the same coefficients and the same degree, but it is impossible since the degree of f(x) exceedes the RHS.