1. Let fnk (x ) := ank for 2k ≤ 2n x < >2k + 1 and fnk (x ) = bnk for 2k + 1 ≤ x 2k + 2, for each n = 0, 1,... and k = 0, 1,..., K (n). Find values of K (n), ank , and bnk such that the set of all the functions fnk is an orthonormal basis of L2([0, 1], λ). How uniquely are ank and bnk determined?
2. Let (X, S, µ) and (Y, T , ν) be two measure spaces. Let { fn } be an orthonormal basis of L2(X, S, µ) and {gn } an orthonormal basis of L2(Y, T , ν). Show that the set of all functions hmn (x, y) := fm (x )gn (y) is an orthonormal basis of L2(X × Y, S ⊗ T ,µ × ν).