Math 641-600 Current Assignment
Assignment 6 - Due Wednesday, December 2, 2009.
- Do the following problems.
- 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.)
- Prove Proposition 0.1
in Notes
on Daubechies' Wavelets
- Prove the polarization identity for u,v in a Hilbert space H:
||u+v||2 + ||u-v||2 = 2(||u||2 +
||v||2).
- Let M be a subspace of a Hilbert space H. Show that M is closed
if and only if M = (M&perp)&perp.
- Let V be a Banach space. Show that a linear operator L:V → V
is bounded if and only if L is continuous.
- 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.
- P is a bounded linear operator, with ||P||=1.
- P2 = P.
- Range(P) = M and Null(P) = M⊥.
- P is self-adjoint, i.e. P* = P.
- 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.
- Show that in the inner product above, H is a Hilbert space.
- Show that the BVP has a unique weak solution in H.
- Section 3.2 problem 3(d), page 128. (Assume the appropriate
operators are closed and that λ is real.)
- Section 3.3 problem 2, page 129. (Assume the appropriate
operators are closed and that λ is real.)
Updated 11/20/09.