MTH 472 "Functional Analysis"

Course Number
 Time
 Location
 Instructor
 Office Hours
 E-mail
67406 MWF 10-10:50 am 1101 Hylan Building Dan-Andrei Geba MW 9-10 am, 806 Hylan Building dangeba@math.rochester.edu

Syllabus: L^p spaces. Normed spaces. Inner product spaces, Hilbert spaces. Linear operators. Duality and the Hahn-Banach theorem. Linear operators on Hilbert spaces. Compact operators. Applications to integral and differential equations.

Prerequisites: Linear algebra, metric spaces, and Lebesgue theory.

Textbook: Bryan P. Rynne and Martin A. Youngson, "Linear Functional Analysis" (2nd Edition), Springer, 2008.

Extra reading material:

All these books will be on reserve at the Carlson Library reserve collection.

Course philosophy:

This is a first course in functional analysis, which builds on the material covered in one year of real analysis. It will contain all the information, pertinent to the subject, that is considered to be essential for the education of a graduate student in Mathematics. We will continue to work on improving your independent reading skills, as reading assignments will be assigned on a weekly basis.

This course is challenging, and, especially in the beginning, it will move quite fast. It will require time commitment. Proficiency will be achieved only by hard work and MASSIVE PROBLEM SOLVING. Please take full advantage of my office hours.

Grading: HOMEWORK 30%, MIDTERM 30%, FINAL 40%.

There are NO MAKE-UP EXAMS. Incomplete "I" grades are almost never given. The only justification is a documented serious medical problem or a genuine personal/family emergency. Falling behind in this course or problems with workload on other courses are not acceptable reasons.

Exams:

The MIDTERM EXAM will be a take-home exam. You will receive it in class on Friday, 10/23, and you will need to hand it in on Monday, also in class, 10/26.

The FINAL EXAM will be on Thursday, 12/17, 12:30-3:30 pm. It will be mostly on the material covered after the midterm, though there will be problems on material pertinent to the midterm.

Homework:

Homework is assigned every other week on Friday, starting 9/11. It will be due in two weeks after it was assigned. The last homework, assigned on 12/4, will need to be turned in at the final exam on 12/17. Late homework is not accepted.

Tentative weekly schedule:


Week of Topic Reading assignment Homework
8/31 Review of linear algebra, metric spaces, and Lebesgue theory
9/7 L^p spaces Sections 2.1-2.2, pages 31-44 Homework 1 (due 9/25).
9/14 Normed spaces, inner product spaces Sections 3.2-3.3, pages 60-71
9/21 Inner product spaces, Hilbert spaces Theorem 3.52, Section 3.5, pages 80-85 Homework 2 (due 10/9).
9/28 Hilbert spaces, linear operators on normed spaces Sections 4.2-4.3, pages 96-108
10/5 Linear operators on normed spaces (part 2) Homework 3 (due 10/30).
10/12 Fundamental theorems related to Banach spaces (i.e., principle of uniform boundedness, open mapping theorem, closed graph theorem)
10/19 Dual of representative spaces (e.g., finite dimensional, Hilbert, l^p spaces) Sections 5.2-5.4, pages 127-144 Take-home Midterm Exam (due 10/26), Homework 4 (due 11/9).
10/26 The Hahn-Banach theorem and its consequences Section 5.7, pages 159-164
11/2 Second dual, dual operators, weak and weak-* convergence Homework 5 (due 11/20): 5.1-5.3, 5.6-5.8, 5.11-5.13, 5.15, 5.19, 5.21, 5.26, 5.28, 5.31.
11/9 Linear operators on Hilbert spaces
11/16 Spectral theory and positive operators, compact operators Section 7.2, pages 216-224 Homework 6 (due 12/4).
11/23 Compact operators (part 2)
11/30 Applications to integral and differential equations Homework 7 (due 12/17).
12/7 Review for the Final Exam STUDY FOR THE FINAL!