Math 641-600 Final Exam Review (Fall 2010)

The final exam will be given on Monday, Dec. 13, from 10:30 am to 12:30 pm, in our usual classroom. It will cover sections 3.2 - 3.6, 4.1 - 4.3.2, 4.5 (pp. 161-163), and it will also cover the material done in class on the Lax-Milgram theorem and Cea's lemma (see class notes for 11/5/10). The test will consist of the following: statements and/or proofs or sketches of proofs of theorems; statements of definitions; proofs of short propositions or solutions of problems similar to ones done in the homework or in class.

Operators and integral equations

Bounded operators
- Norms of linear operators, unbounded operators, continuous linear functionals, spaces associated with operators
- Integral equations -- Hilbert-Schmidt kernels, Fredholm, Volterra
- The Projection (Decomposition) Theorem (see your notes for 10/27/10)
- The Riesz Representation Theorem
- Existence of adjoints of bounded operators
- Lax-Milgram Theorem and Cea's Lemma (see your notes for 11/5/10)
Compact operators
- Finite rank operators, Approximation Theorem (Theorem 3.4), and Hilbert-Schmidt kernels/operators
- Spectral theory for compact, self-adjoint operators, K = K*
  - Eigenvalues and eigenspaces
    - Eigenvalues are real; eigenvectors for distinct eigenvalues are orthogonal
    - Eigenspaces are finite dimensional
    - The only limit point of the set of eigenvalues is 0.
    - "Maximum principle" (p. 117)
  - Completeness of eigenfunctions on the closure of the range of K (Theorem 3.6)
  - Application to eigenfunction problems involving integral equations
- Closed Range Theorem (Theorem 3.7), Fredholm alternative, resolvents and kernels
- Contraction Mapping Theorem, Neumann series, solving integral equations

Distributions and applications

Test function space D, distribution space D′, examples, δ function, integral representation, derivatives of distributions
Green's functions

Differential operators

Unbounded operators, domain of an operator, adjoints
Eigenfunction expansions

Additional Problems (Chapter 4)

Section 4.1: 5, 8
Section 4.2: 2, 4
Section 4.3: 2, 3, 5
Let Lu = − (xu′ )′, D(L) := {u, Lu ∈ L²[1,2], u(1) = u(2) = 0}.
1. Show that L is self adjoint and positive definite with respect to the inner product for L²[1,2].
2. Find the Green's function for L.
3. Directly — i.e., without quoting Theorem 4.7 — use the Green's function to show that the eigenfunctions of L are complete in L²[1,2].

Updated 12/5/2010 (fjn).