Instructors:
Todd Arbogast
Office: RLM 11.162 (phone 471-0166) and ACE 5.334 (phone 475-8628)
E-Mail: arbogast@ticam.utexas.edu

Irene Gamba
Office: RLM 10.166 (phone 471-7150) and ACE 3.340 (phone 471-7422)
E-Mail: gamba@math.utexas.edu

Office Hours (for Irene Gamba): Thursdays 1:00-2:00pm and by appointment.

Meeting: MWF 12:00-1:00, RLM 12.166, and a weekly problem discussion session will be scheduled.

Teaching Assistant: Diego Moreira. Office: RLM XXX (phone 471-XXXX)
E-Mail: dmoreira@math.utexas.edu

Course Description: This is the first semester of a course on methods of applied mathematics. It is open to mathematics, science, engineering, and finance students. It is suitable to prepare graduate students for the Applied Mathematics Preliminary Exam in mathematics and the Area A Preliminary Exam in CAM. The first semester is an introduction to functional analysis.

Textbook: E. Kreyszig, Introductory Functional Analysis with Applications, Wiley, 1978. Lecturer-prepared notes. A recommended supplemental text are the T.Arbogast and J. Bona lecture--notes to be available in class and at Lecturer-prepared notes

Additional bibliography:

1. L. Debnath & P. Mikusinski, Introduction to Hilbert Spaces with Applications, Acad. Press, 1990.
2. J.T. Oden & L.F. Demkowicz, Applied Functional Analysis, CRC Press, 1996.
3. M. Reed & B. Simon, Methods of Modern Physics, Vol. 1, Functional analysis.
4. H.L. Royden, Real analysis, 3rd ed., Macmillan, 1988.
5. W. Rudin, Functional Analysis, 1991.
6. W. Rudin, Real and Complex Analysis, 3rd Ed., 1987.
7. K. Yosida, Functional Analysis, Springer-Verlag, 1980.
8. E.W. Cheney and H.A. Koch, Notes on Applied Mathematics, Department of Mathematics, University of Texas at Austin.

Homework, Exams, and Grades: Homework will be assigned regularly. Students are encouraged to work in groups; however, each student must write up his or her own work. Two mid-term exams will be given. The final exam will be given during finals week (officially scheduled for Wednesday, December 8, 2:00-5:00pm). The final grade will be based on the homework grade and the three exam grades.

Sillabus

Semester I.

0. Preliminaries (3 weeks)

1. 1. Metric spaces and introductory topology.
2. 2. Lebesgue measure and integration.
3. 3. Lebesgue spaces.

1. Banach Spaces (5 weeks)

1. 1. Normed linear spaces and convexity.
2. 2. Convergence, completeness, and Banach spaces.
3. 3. Continuity, open sets, and closed sets.
4. 4. Continuous Linear Transformations.
5. 5. Hahn-Banach Extension Theorem.
6. 6. Linear functionals, dual and reflexive spaces, and weak convergence.
7. 7. The Baire Theorem and uniform boundedness.
8. 8. Open Mapping and Closed Graph Theorems.
9. 9. Closed Range Theorem.
10. 10. Compact sets and Ascoli-Arzela Theorem.
11. 11. Compact operators and the Fredholm alternative.

2. Hilbert Spaces (3 weeks)

1. 1. Basic geometry, orthogonality, bases, projections, and examples.
2. 2. Bessel's inequality and the Parseval Theorem.
3. 3. The Riesz Representation Theorem.
4. 4. Compact and Hilbert-Schmidt operators.
5. 5. Spectral theory for compact, self-adjoint and normal operators.
6. 6. Sturm-Liouville Theory.

3. Distributions (3 weeks)

1. 1. Seminorms and locally convex spaces.
2. 2. Test functions and distributions.
3. 3. Calculus with distributions.

Semester II.

1. The Fourier Transform and Sobolev Spaces

2. Variational Boundary Value Problems (BVP)

3. Differential Calculus in Banach Spaces and Calculus of Variations

4. Asymptotic Analysis