MATH632-0101: Functional Analysis-Fall 2022

Meeting times: MWF, 9:00am-9:50am (PHY 2214)

Instructor: Professor Jonathan Rosenberg. His office is room 2114 of the Math Building, phone extension 55166, or you can contact him by email. His office hours are M and W 10-11, or by appointment.  The course TA (for grading homework) is ... (TBA).

Text: Peter Lax, Functional Analysis (Wiley). You can get either the hardcover or the ebook. The book is available at a substantial discount at amazon.com and possibly other online bookstores, so shop around.

Prerequisite: MATH 631 (real analysis).  Actually, somewhat less real analysis may do if you have a good command of basic linear algebra and have taken MATH 630 (first semester real analysis) or equivalent.

Catalog description:

Introduction to functional analysis and operator theory: normed linear spaces, basic principles of functional analysis, bounded linear operators on Hilbert spaces, spectral theory of self-adjoint operators, applications to differential and integral equations, additional topics as time permits.

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Course Description:

Lax's book has a lot more in it than can be covered in one semester, but it's readable and gives you a very good feeling for what the subject is good for. I am planning to concentrate on the following topics:

• The Hahn-Banach Theorem in all its various forms (Chapters 3-4). This subject reoccurs in another incarnation in Chapter 8.
• Banach spaces and Hilbert spaces (Chapters 5-7). It is especially important to have a good feel for the geometry of the latter.
• Weak compactness, a fundamental tool (Chapter 12)
• Convex sets and the Krein-Milman Theorem (Chapters 13-14)
• Uniform Boundedness and the Closed Graph Theorem (Chapter 15)
• Basics of Banach Algebras, especially commutative ones (Chapters 17-19)
• Spectral theory of self-adjoint, and especially self-adjoint compact, operators on Hilbert space (Chapters 22-23, 28-29, 31)

If you can, try to look at the "applications" chapters, even if we don't have time to cover them in class. They are some of the best parts of the book, and show you the power of the subject.

Files Tab:

Please remember to look at the Files tab in this website.  In it you will find some of the original classic papers in the subject (always instructive to read), as well as things like homework solutions.

Course Requirements and Grading Policy:

Homework will be assigned, collected, and graded, usually once a week. Homework counts for 70% of the grade. There will also be a take-home final exam due Saturday, December 17, at 10AM (the exam time for the course in the official exam schedule), counting for 30% of the grade. Grades will have the following rough meaning (modified by pluses and minuses, as appropriate):

• A. Did most of the problems correctly, seemed to have a good idea of what's going on.
• B. Did a fair number of problems, but had some trouble with them.
• C. Didn't do much of the homework, or got essentially none of it right.
• D, F. I hope these won't arise. Basically mean "hardly ever showed up, no effort at all".

It is your responsibility to turn the homework and the final exam in on time.

Attendance Policy, Academic Dishonesty, Disabilities:

I won't take attendance, but missing a lot of classes might affect your grade if you are near the borderline between two grades.  On the homework, it is OK to talk to your fellow students about the problems (in this regard you can use the Chat feature on the website), but you should write up the answers yourself.  On the take-home final exam, no collaboration is allowed, and you are expected to write out and sign the honor pledge:

I pledge on my honor that I have not given or received any unauthorized assistance on this assignment/examination. 

If you think you need accommodation for some disability, please talk to the instructor and we'll make suitable arrangements.