2. Let X be an inner product space and e1, C2, ..., en for some n e N be pairwise orthogonal elements. Prove Pythagoras' Theorem: ΙΣe;l? = Σle;l? n n j=1 j=1 3. Let X := Cº[-1, 1] = {f:(-1, 1] C | f is continuous}. Show that (:-): X X X+C given by (f,g) := L i f(t)g(t) dt is a complex inner product. 4. On CR[-1, 1] = {f: [-1, 1] + R | f is continuous}, considered with its usual inner product as in 3, show that the set of odd functions is a closed subspace. Then find the orthogonal complement of it. Give arguments. 5. Consider L2R(R) with the inner product (f,g) Low(t)f(t)g(t) dt where w(t) := e. Find an orthonormal system {€0, 61, 62} whose span is the same as the polynomials 1, t, t2. Hint. Use the Gram - Schmidt orthonormalization process.