## Math 641-600 Final Exam Review (Fall 2017)

The final exam will be given on Monday, Dec. 11, 10:30 am-12:30 pm, in
our usual classroom. It will cover sections 2.2.7, 3.2 - 3.6, 4.1 -
4.3.2, 4.5, and all class notes, starting from Splines and finite
element spaces, except for the notes on X-ray tomography. 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. Extra office hours: Wednesday (12/6)
1-3, Thursday, (12/7) 10-12, Friday (12/8), 1-3. For other times, send
me an email to arrange an appointment.
### Finite elements

### Operators and integral equations

- Bounded operators (§3.2, Bounded
Operators & Closed Subspaces,
and
Projection theorem, the Riesz representation theorem, etc.)
- Norms of linear operators, unbounded operators, continuous linear
functionals, spaces associated with operators
- Hilbert-Schmidt kernels
- The Projection Theorem
- The Riesz Representation Theorem
- Existence of adjoints of bounded operators
- Fredholm alternative

- Compact operators (§3.3, §3.5, Compact
Operators and on
Closed Range Theorem.)
- Finite rank operators, $\mathcal C(\mathcal H)$ is a closed
subspace of $\mathcal B(\mathcal H)$ (Keener, Theorem 3.4 and Compact
Operators, Theorem 2.6) and Hilbert-Schmidt kernels/operators
- Closed Range Theorem, Fredholm alternative, resolvents and
kernels

- Spectral theory for compact, self-adjoint operators, K = K*
(§3.4,
Spectral Theory for Compact Operators.)
- 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" (Keener, p. 117 and Spectral
Theory for Compact Operators, Lemma 2.5)

- Completeness of eigenfunctions on the closure of the range of K
(Kener, Theorem 3.6 and Spectral
Theory for Compact Operators, Theorem 2.7)
- Application to eigenfunction problems involving integral equations

- Contraction Mapping Theorem, Neumann series (§3.6;)

### Distributions and differential operators

- Test function space $\mathcal D$, distribution space $\mathcal D'$, integral
representation; delta functions, derivatives of
distributions (§4.1, Example
problems on distributions.)
- Green's functions for 2nd order operators (§4.2)
- Domain of an operator, adjoints of 2nd order operators, domain of
the adjoint, self-adjoint differential operators (§4.3)
- Completeness of the set of o.n. eigenfunctions for a self-adjoint
differential operator (class notes, 12/6/17 and Keener, §4.5)

Updated 12/7/2017 (fjn).