1 Prove that the following are vector spaces over the reals. (i) The space F(X) of all functions f : X R from an arbitrary set X (not necessarily a vector space!) into R, with addition and scalar multiplication defined pointwise, i.e., (f g)(x) := f(x) g(x) and (f)(x) := f(x) for any two f, g : X R and for any R. (ii) The space of rational functions of a single complex variable s, i.e., those functions h : C C that can be expressed in the form p(s)/q(s), where p and q are polynomials with real coefficients and q is not the zero polynomial. Addition and scalar multiplication are defined pointwise. (iii) For two vector spaces X and Y over the reals, the space L(X, Y) of all linear maps from X to Y, i.e., T L(X, Y) iff the following holds for all x, x0 X and all , R: T(x x0 ) = T(x) T(x 0 )