MTH 614: Functional Analysis, Winter, 2007
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.
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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)
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