Math 641-600 Current Assignment

Assignment 6 - Due Monday, November 30, 2009.

• Do the following problems.
  1. Section 2.2: 26. (The definition of projection applies to to Banach spaces as well as Hilbert spaces. The two parts of the problem involve Banach spaces.)

  2. Prove Proposition 0.1 in Notes on Daubechies' Wavelets

  3. Prove the polarization identity for u,v in a Hilbert space H:
     ||u+v||² + ||u-v||² = 2(||u||² + ||v||²).
     

  4. Let M be a subspace of a Hilbert space H. Show that M is closed if and only if M = (M^&perp)^&perp.

  5. Let V be a Banach space. Show that a linear operator L:V → V is bounded if and only if L is continuous.

  6. Let M be a closed subspace of a Hilbert space H. Let h be in H, and let p be the unique minimizer of || h - u|| over all u in M. Define the operator P:H → H by Ph = p. The operator P is called the projection of H onto M. Show that the following are true.
     1. P is a bounded linear operator, with ||P||=1.
     2. P² = P.
     3. Range(P) = M and Null(P) = M^⊥.
     4. P is self-adjoint, i.e. P^* = P.
  7. Consider space H = comprising all functions f in Sobolev space H¹[0,1] that satsify f(0) = f(1) = 0. Let v(v) be continuous and strictly positive on [0,1]. On H, define the inner product
     < f,g >_H = ∫₀¹(f ′(x) g ′(x) +v(x)f(x)g(x))dx,
     A weak solution u to the boundary value problem (BVP)

     -u''+v(x)u = h(x), u(0) = u(1) = 0, h ∈ L²[0,1].
     is a function u ∈ H such that for all f ∈ H we have < u, f >_H = ∫₀¹h(x)f(x)dx.

     1. Show that in the inner product above, H is a Hilbert space.
     2. Show that the BVP has a unique weak solution in H.

  8. Section 3.2 problem 3(d), page 128. (Assume the appropriate operators are closed and that λ is real.)

  9. Section 3.3 problem 2, page 129. (Assume the appropriate operators are closed and that λ is real.)

Updated 11/20/09.