MTH 614: Functional Analysis, Fall, 2012
This two-term 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, the Closed Range Theorem,
and the Uniform Boundedness Principle in Banach space. Additional
topics include variational principles for minima or saddle points,
convex analysis, Lp and Sobolev spaces and corresponding trace spaces
of boundary values, semigroup theory, and applications to partial
differential equations.
Winter Term is offered as MTH 619: Topics in Analysis.
Prerequisite:
6 credits of senior-level analysis.
Special agreements will be arranged with students without formal
prerequisites, particularly those interested primarily in the applications.
Textbook: Functional Analysis, Sobolev Spaces and Partial
Differential Equations, Haim Brezis, Springer, 2011.
Additional materials will be provided for examples, variations and
applications. |
Final Exam: Thursday, 6 December 1400
NOTES
Schedule:
1. I.1 Linear algebra pp. 1-5. NOTE: Use \tilde(G) to
denote `restrictions to \bar(G)' on top of p.3. (9/24)
2. I.2 Convergence and continuity pp. 5-9.
3. Examples.
4. I.3 Completeness pp. 10-14. (10/01)
5. I.4.1 Hilbert Space,
Minimization principle , and Text: 5.1 (pp 131-134).
6. I.4.2-4 Projection and Dual, and Text: 5.2.
7. I.5 dual operators, identification and Text 5.2. (10/08)
8. I.7 Fourier series pp. 24-25 and Text: 5.4.
9. Expansion in Eigenfunctions pp. 26-27 and Text: 6.4.
10. Baire category theorem Text: 2.1 and I.6. (10/15)
11. Uniform boundedness; weak compactness Text: 2.2.
12. Open mapping, closed graph, equivalent norms Text: 2.3.
13. Zorn, well ordering & axiom of choice (10/22)
14. Hahn-Banach extension theorem Text: 1.1.
15. dual operators
16. annihilator of the range (10/29)
17. surjective operators Text: 2.7 & 1.2.
18. closed range theorem,
Exercises due 11/12
19. Hahn-Banach separation theorems Text: 1.2. (11/05)
20. Variational Method: preview, pp. 1-6. See NOTES above.
21. H^1(a,b), pp. 7-10.
22. boundary-value problems, 1D, pp. 15-18. Text: 8.1-8.4. (11/12)
23. unilateral constraints
24. Lax-Milgram theorem Text: p. 140.
25. Stampacchia's theorem (11/19)
26.
27. Holiday
28. eigenvalue problems pp. 28-32, Text: pp. 231-233.(11/26)
29. characterization of subspaces, interpolation pp. 33-34.
30. evolution problems pp. 35-37.
Exercises due 12/6
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