1. The problem statement, all variables and given/known data 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. 3. The attempt at a solution Well, I got part 1. As for part 2... I have no clue. Any ideas?