17.7. EXERCISES 423

26. Let A be an n×n matrix such that A = A∗. Verify that ⟨Ax,x⟩ is always real. Then Ais said to be nonnegative if ⟨Ax,x⟩ ≥ 0 for every x. Verify that [·, ·] given by [x,y]≡⟨Ax,y⟩ satisfies all the axioms of an inner product except for the one which says that[x,x] = 0 if and only if x = 0. Also verify that the Cauchy Schwarz inequality holdsfor [·, ·].

27. Verify that for V equal to the space of continuous complex valued functions definedon an interval [a,b] , an inner product is

⟨ f ,g⟩=∫ b

af (x) ḡ(x)dx

If the functions are only assumed to be Riemann integrable, why is this no longer aninner product? In this last case, does the Cauchy Schwarz inequality still hold?