Sorry for the vague title!
Let R denote the set of real numbers, and F(S,R) denote the set of all functions from a set S to R.
Part 1: Let [tex]\phi[/tex] be any mapping from a set A to a set B. Show that composition by [tex]\phi[/tex] is a linear mapping from F(B,R) to F(A,R). That is, show that [tex] T : F(B,R) \rightarrow F(A,R) : f \mapsto f \circ \phi[/tex] is linear.
Part 2: In the situation given in part 1, show that T is an isomorphism if [tex]\phi[/tex] is bijective by show that:
(a) [tex]\phi[/tex] injective implies T surjective;
(b) [tex]\phi[/tex] surjective implies T injective.
The Attempt at a Solution
Well, I got part 1.
As for part 2... I have no clue. Any ideas?