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: Open office hours: unless the instructors have
pressing business, they will be available to help students who come
by. Students may also make specific appointments for help.
Meeting: MWF 12:00-1:00, RLM 11.176, and a weekly problem
discussion session will be scheduled.
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: Lecturer-prepared notes.
A recommended supplemental text is E. Kreyszig, Introductory
Functional Analysis with Applications, Wiley, 1978.
Bibliography:
- L. Debnath & P. Mikusinski, Introduction to Hilbert Spaces
with Applications, Acad. Press, 1990.
- E. Kreyszig, Introductory Functional Analysis with
Applications, 1978.
- J.T. Oden & L.F. Demkowicz, Applied Functional Analysis, CRC
Press, 1996.
- M. Reed & B. Simon, Methods of Modern Physics, Vol. 1,
Functional analysis.
- H.L. Royden, Real analysis, 3rd ed., Macmillan, 1988.
- W. Rudin, Functional Analysis, 1991.
- W. Rudin, Real and Complex Analysis, 3rd Ed., 1987.
- K. Yosida, Functional Analysis, Springer-Verlag, 1980.
- 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 Tuesday, December 19, 9:00-12:00 noon). The final grade
will
be based on the homework grade and the three exam grades.
Semester I.
0. Preliminaries (3 weeks)
- 1. Topological spaces and metric spaces.
- 2. Lebesgue measure and integration.
- 3. Lebesgue spaces.
1. Banach Spaces (5 weeks)
- 1. Normed linear spaces and convexity.
- 2. Convergence, completeness, and Banach spaces.
- 3. Continuity, open sets, and closed sets.
- 4. Continuous Linear Transformations.
- 5. Hahn-Banach Extension Theorem.
- 6. Linear functionals, dual and reflexive spaces, and weak
convergence.
- 7. The Baire Theorem and uniform boundedness.
- 8. Open Mapping and Closed Graph Theorems.
- 9. Closed Range Theorem.
- 10. Compact sets and Ascoli-Arzela Theorem.
- 11. Compact operators and the Fredholm alternative.
2. Hilbert Spaces (3 weeks)
- 1. Basic geometry, orthogonality, bases, projections, and examples.
- 2. Bessel's inequality and the Parseval Theorem.
- 3. The Riesz Representation Theorem.
- 4. Compact and Hilbert-Schmidt operators.
- 5. Spectral theory for compact, self-adjoint and normal operators.
- 6. Sturm-Liouville Theory.
3. Distributions (3 weeks)
- 1. Seminorms and locally convex spaces.
- 2. Test functions and distributions.
- 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