Math 617 Functional Analysis

Spring 2020, UAF

UPDATED SYLLABUS


SEE THE ORIGINAL SYLLABUS HERE

Instructor:
   Ed Bueler
   Chapman 306C
   office hours:   bueler.github.io/OffHrs.htm
   elbueler@alaska.edu 		Class:
   MWF 10:30 -- 11:30 am
   Chapman 204 ONLINE!
   CRN: 35419

Textbook:    J. Muscat, Functional Analysis: An Introduction to Metric Spaces, Hilbert Spaces, and Banach Algebras, Springer, 2014

Course Topics:

UAF catalog description: Study of Banach and Hilbert spaces, and continuous linear maps between them. Linear functionals and the Hahn-Banach theorem. Applications of the Baire Category theorem. Compact operators, self adjoint operators, and their spectral properties. Weak topology and its applications.

The above description is reasonably accurate, but I will emphasize connections with finite-dimensional linear algebra and with the axioms of quantum mechanics. I will emphasize the spectral theory of compact and bounded self-adjoint operators.


ONLINE VIDEO LECTURES

Instead of in-person lectures, starting 23 March there are video lectures available at the Blackboard site. The goal is to have less than one hour of video lecture per originally-scheduled hour.

Goals and Outcomes:

Why this course? Here are three reasons:
  1. Functional analysis is an important part of 20th-century mathematics, at the intersection of linear algebra, topology, and real analysis, with applications to differential geometry, probability, and Lie groups, so it is an important course in pure mathematics.
  2. Functional analysis is the fundamental technology for understanding partial differential equations (PDEs), for instance in the Sobolev space theory of elliptic PDEs and their finite-element analysis, or in the distributional and Fourier-transform theory of the wave, heat, and Schrodinger equations, so it is an important course for research in applied mathematics and numerical analysis.
  3. The mathematical theory of quantum mechanics, not to mention quantum field theory, is mostly functional analysis.

Course Website:

bueler.github.io/M617S20/

The site also has a daily schedule of topics which will be updated on an ongoing basis to reflect which topics were actually covered. Due dates for homework, and the dates of exams, will be in this schedule.

Homework:

Weekly homework forms 40% of your score for the course. It consists mostly of proofs, but there will also be computations of examples and counter-examples, and some visualization (sketches). You are encouraged to work with other students, and seek help from the instructor, on homework problems. However, the work you turn in must be your own.

Homework will be due at the beginning of class on the announced date, and late homework will not be accepted. Homework assignments and their due dates will regularly be posted at the above website.

On each assignment, one problem for each student will be identified as that student's "LaTeX problem". This must be sent electronically to me by the due date. I will suggest revisions immediately. Then an edited version must be sent within 24 hours. The edited version will then be included in the solution set which, once complete, will be emailed directly to students. See the website for LaTeX help and materials.

Exams: (UPDATED)

There will be two Midterm Exams, the first in-class for a full hour and the second take-home. These will generally cover recent material. The in-class, two hour TAKE-HOME Final Exam will have problems from the whole semester.

How Your Grade is Determined:

Portion
Homework
Midterm Exam 1
Midterm Exam 2 (take-home)
Final Exam (TAKE-HOME)
 Percentage
40%
15%
15%
30%
Dates
weekly
in class Monday 2 March
assigned Monday, 6 April
in class Thursday, 30 April, 10:15 am -- 12:15 pm
assigned/due Friday 24 April/Friday 1 May

A numerical score will be assigned for each homework and exam, and they will be weighted as above. Based on your raw score total, I will assign grades according to the following schedule:

   90 - 100 % = A,  79 - 89 % = B,  68 - 78 % = C,  57 - 67 % = D,  0 - 56 % = F.

This schedule is a guarantee. I reserve the right to increase your grade above these ranges based on the actual difficulty of the work and/or upon average class performance.

Policies:

The Dept of Mathematics and Statistics has reasonable policies on incompletes, late withdrawals, early final examinations, etc.; see www.uaf.edu/dms/policies.   You are covered by the UAF Student Code of Conduct.   I will work with the Office of Disabilities Services (208 WHIT, 474-5655) to provide reasonable accommodation to students with disabilities.

Prerequisites:

The absolute prerequisites are an undergraduate course in linear algebra (e.g. MATH 314) and an undergraduate, proof-based course in real analysis (e.g. MATH 401). Exposure to additional mathematics is, of course, recommended, including undergraduate topology and complex analysis, and graduate real analysis (e.g. MATH 641).