Linear algebra question. Please show all work.

Let C = C[-1, 1] be the vector space of all continuous real valued functions on the interval [-1, 1]. A function f in C is even if f(-x) = f(x) for all x E [-1, 1]; it is odd if f(-x) = -f(x) for all xe [-1, 1]. Let Co = {fe C : f is odd} and Ce = {f E C : f is even}. 1. Prove that Co and Ce are both subspaces of C. 2. Prove that Co and Ce are orthogonal subspaces, meaning that any f E C. and g E Ce must satisfy (f, g) = f(x)g(x) dx = 0