Math 641-600 Final Exam Review (Fall 2021)
The final exam will be given on Tuesday, Dec. 14, 8-10 am, in our
usual classroom. The exam itself will have 4 to 6 questions, some
with multiple parts. 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. Additional office hours: Friday
(12/10), 11:30-12:30; Monday (12/13), 11-12:30 and 2-3. For other
times, send me an email to arrange an appointment. (The office hours
are subject to change. If I need to do that, I'll email the class.)
Finite Elements
Operators and Spectral Theory
- 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.)
- $\mathcal C(\mathcal H)$ is a closed subspace of $\mathcal
B(\mathcal H)$ (Keener, Theorem 3.4 and
Compact Operators, Theorem 2.6)
- Hilbert-Schmidt kernels/operators are compact.
- 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 for $K$.
- Eigenvalues are real; eigenvectors for distinct eigenvalues are
orthogonal; eigenspaces are finite dimensional
- The only limit point of the set of eigenvalues is 0; the
spectrum, $\sigma(K)$ consists of $0$, which may or may not be an
eigenvalue, and eigenvalues that are real, discrete and have 0 as
their only limit point.
- "Maximum principle" (Keener, p. 117 and Spectral
Theory for Compact Operators, Lemma 2.5)
- The Spectral Theorem (Keener, Theorem 3.6 and Spectral
Theory for Compact Operators, Theorem 2.7)
- Solving eigenvalue problems
- Contraction Mapping Theorem, Neumann series (Keener, §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 (Sturm-Liouville) operator. (Class notes and Keener, §4.5)
- Be able to state the Courant-Fischer Theorem for a
Sturm-Liouville eigenvalue problem, and briefly (1/2 page,
max) sketch its proof.
Updated 12/3/2021 (fjn).