METHODS OF APPLIED MATHEMATICS I



MATH 383C (Unique # 58050) and CAM 385C (Unique # 66065)

FALL SEMESTER 2009

Instructor

Thomas Chen
Office: RLM 12.138, Phone: 471 7180
E-Mail: t c A_T m a t h . u t e x a s . e d u
Office Hours: Mo, 1-2 PM (individual) and Fr, 1-2 PM (group, in ENS 115).

Teaching Assistant

Timothy Blass
Office: 12.144, Phone: 475 8689
E-Mail: t b l a s s A_T m a t h . u t e x a s

Time and Location

Classes meet on MWF 12:00 - 12:50 PM, in RLM 7.104.

Problem discussion sessions Fridays 1:00 - 1:50 PM, in ENS 115.

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.

Grading

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. The final grade will be based on the homework grade and the three exam grades.


MIDTERM 1 on Friday, October 9, 2009. Location: RLM 4.102. Time: 4:00 - 5:30 PM.

MIDTERM 2 on Friday, November 13, 2009. Location: BUR 112. Time: 4:00 - 5:30 PM.

FINAL EXAM on Monday, December 14, 2009. Location: RLM 7.104. Time: 9:00 - 12:00 AM.

Textbook

We will use the lecture notes by T. Arbogast and J. Bona.
Please purchase the updated version at the UT Copy Center in Welch 2.228 (many thanks to Prof. Arbogast for making it available).

A recommended supplemental text is E. Kreyszig, Introductory Functional Analysis with Applications, Wiley, 1978.

Homework Assignments

The homework assignments will be posted here.
  1. HW 0, for August 26: Please read chapter 1 in the lecture notes (topological and metric spaces).
  2. HW 1, hand-in on September 4.
  3. HW 2, hand-in on September 11.
    Note: For problem 4(b), you may invoke the Lebesgue monotone convergence theorem. The union in problem 2 is countable.
  4. HW 3, hand-in on September 18.
    Note: Section 2, problem 24 added on Mo, Sep 14 (was not contained in previous version due to file transfer error).
  5. HW 4, hand-in on September 25.
    Note: Misprint on first version: Section 2, problem 22 should be problem 25. This has been corrected in the present version.
    Here is a very nice discussion of Zorn's lemma, by W.T. Gowers (with comments by T. Tao).
  6. HW 5, hand-in on October 2.
    Note: Section 2, problem 46 has been changed to problem 41, on 9/30/09.
    Note: The solutions posted on blackboard have been updated (omissions in # 32, 33 fixed), on 10/6/09, 10:20 PM.
  7. HW 6, hand-in on October 16.
  8. HW 7, hand-in on October 23.
    Note: Typo in Problem 3 fixed (the second factor in the sum depends on y, not x), on 10/17/09, 10:40 PM.
  9. HW 8, hand-in on Mo, November 2.
    Note: Problem 6: X is Banach, added on 10/26/09, 8:15 PM. Handin date moved on 10/29/09, 9:40 AM.
  10. HW 9, hand-in on Fr, November 6.
  11. HW 10, hand-in on Fr, November 20.
  12. HW 11, hand-in on Mo, November 30.
Link to Blackboard.

Approximate schedule

1. Preliminaries (2 weeks)
    a. Topological spaces and metric spaces.
    b. Lebesgue measure and integration.
    c. Lebesgue spaces.
2. Banach Spaces (5 weeks)
    a. Normed linear spaces and convexity.
    b. Convergence, completeness, and Banach spaces.
    c. Continuity, open sets, and closed sets.
    d. Continuous Linear Transformations.
    e. Hahn-Banach Extension Theorem.
    f. Linear functionals, dual and reflexive spaces, and weak convergence.
    g. The Baire Theorem and uniform boundedness.
    h. Open Mapping and Closed Graph Theorems.
    i. Closed Range Theorem.
    j. Compact sets and Arzela-Ascoli Theorem.
    k. Compact operators and the Fredholm alternative.
3. Hilbert Spaces (4 weeks)
    a. Basic geometry, orthogonality, bases, projections, and examples.
    b. Bessel's inequality and the Parseval Theorem.
    c. The Riesz Representation Theorem.
    d. Compact and Hilbert-Schmidt operators.
    e. Spectral theory for compact, self-adjoint and normal operators.
    f. Sturm-Liouville Theory.
4. Distributions (3 weeks)
    a. Seminorms and locally convex spaces.
    b. Test functions and distributions.
    c. Calculus with distributions.

Bibliography

  1. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol 1 (Functional Analysis), Academic Press 1980.
  2. H.L. Royden, Real analysis, 3rd ed., Macmillan, 1988.
  3. W. Rudin, Functional Analysis, 1991.
  4. W. Rudin, Real and Complex Analysis, 3rd Ed., 1987.
  5. K. Yosida, Functional Analysis, Springer-Verlag, 1980.
  6. E.W. Cheney and H.A. Koch, Notes on Applied Mathematics, Department of Mathematics, University of Texas at Austin.

The University of Texas at Austin provides upon request appropriate academic accommodations for qualified students with disabilities.
For more information, contact the Office of the Dean of Students at 471-6259, 471-4641 TTY.