Prove results similar to Exercises 46 and 47 for the adjoint of a linear operator on a real or complex inner product space.
Exercise 46 & 47
46. Let T : V → W be a linear map. (a) Prove ker(T∗ ) = A(img(T)). (b) Deduce that T∗ is injective iff T is surjective.
47. Let T : V → W be a linear map. (a) Prove img(T∗ ) = A(ker(T)). (b) Deduce that T∗ is surjective iff T is injective