Let C([-1,1]) be the vector space of all continuous functions f on the interval [-1,1]. Define P: C([-1,1]) -> C([-1,1]) to be the function which inputs a continuous function f and outputs its even part Pf given by (Pf)(x) = [f(x) + f(-x) ] / 2. Show that P is linear and that it is a projection, i.e., satisfies P2 = P.
I know that in order to show that this is linear I must show it preserves addition and scalar multiplication, but I'm having trouble going from all continuous functions to just the even part? And how would I show this is a projection?