5. Let Man(C) be the set of all n x n matrices with complex entries. Define the map (): M..(C) x M..(C) + C by (A,B) = tr(B* A). Here A' is the conjugate transpose of A. (a) Prove that (,) defines an inner product on Min (C) and hence it defines a norm on Mon(C) by || A|| = tr(AA). (b) Prove that Msn(C) is a Hilbert space with this inner product. (e) Prove that ||AB||