MTH 614: Functional Analysis, Winter, 2007

LECTURE: MW 1500 - 1615 Batcheller Hall 250 CRN 28875
Instructor: R.E. Showalter Kidder 368 show@math.oregonstate.edu

This course will begin with the elements of Hilbert space, where most basic principles appear in their simplest form. Then we move on to the fundamentals of functional analysis, including the Hahn-Banach Theorem, Open Mapping Theorem, and the Uniform Boundedness Principle in Banach space. Additional topics include variational principles, the Closed Range Theorem, Lp and Sobolev spaces, semigroup theory, and applications to initial and boundary value problems.

Prerequisite: 6 credits of senior-level analysis.

Textbook: All material for the course is available on the web.

This course provides an introduction to basic topics of Hilbert space and variational principles, which are to be used for MTH 657 Topics in Applied Math: Homogenization and Multiscale Problems of Flow and Transport to be offered in Spring, 2007.

Schedule:
Chapter I: Elements of Hilbert Space
1. I.1 Linear algebra pp. 1-5. NOTE: Use \tilde(G) to denote `restrictions to \bar(G)' on top of p.3. (1/8/07)
2. I.2 Convergence and continuity pp. 5-9.
3. I.3 Completeness pp. 9-11.
Exercises due 1/24: pp. 27-28: 1.4, 2.3, 2.6 [watch missprints: T(S) should be T^{-1}(S), and `T is continuous only if K(T) is closed.']
4. I.4 Hilbert space pp. 12-16. (1/22)
5. p. 16 and Minimization principle
6. pp. 17-18. (1/29)
7. pp. 19-20. Exercises due 2/7
8. I.6 Uniform boundedness; weak compactness pp. 22-24. (2/5)
9. I.7 Fourier series pp. 24-25.
10. Expansion in eigenfunctions pp. 26-27. (2/12)
11. Two-Point Boundary-value problems.
Chapter II: Distributions and Sobolev Spaces
12. Lebesgue spaces and derivatives pp. 31-33. (2/19)
13. Derivatives and Sobolev space pp. 34-39.
Exercises due 2/28: p. 55: 1.1, 1.2.
14. Elliptic Boundary-Value Problems (2/28)
15. Category and Uniform Boundedness (3/05)
16. Open Mapping Theorem
17. Hahn-Banach Theorem and Transfinite Induction(3/12)
18. Closed Graph Theorem, Dual Operators, and Closed Range Theorem
Final Exercise : Numbers 1, 2, 3 and one more are due morning of 3/22.