PROBLEM 2: (Versions of the spectral theorem)

(a) If v is an eigenvector of A with eigenvalue
    lambda, what can you say about A2 v?

(b) Write T=A1+i A2, where A1 and A2 are symmetric, 
    compact, and commute.

(c) Gratuitous remark: We must have |gammma| = 1 (why?)


    
PROBLEM 3: (Hilbert-Schmidt operators)

(a) Schwarz' inequality

(b) Approximate K by simple functions



PROBLEM 4: (Weak density of the unit sphere in the ball)

(b) Use Part (a)



PROBLEM 5: (Stereographic projection of S^1)

(a) Change variables x= tan (\theta/2)



PROBLEM 6: (Commuting projections)

Hint: Consider ||P1 P2 x||  for x in H