Course on Functional Analysis Vanderbilt Fall 2020

Syllabus:

Function spaces, topological vector spaces, linear operators, conjugate spaces, Hilbert and Banach spaces, Banach algebras. Applications to function theory, differential equations, and integral equations. Prerequisite: 6100.

Course Description.

Linear algebra review including the index of an operator,the Hahn-Banach theorem, the open mapping theorem, the closed graph theorem and the uniform boundedness principle. Banach spaces, L^p spaces,locally convex topolocical vector spaces, distributions, Hilbert space. Compact operators. Spectral theorem for self-adjoint and normal compact operators. Trace-class and Hilbert-Schmidt operators. Fredholm operators and index revisited. Functional calculus. Banach algebras.General spectral theorem.

Textbook:

The text for the course will be Lax’s “Functional Analysis”

Other textbooks I like.

Reed and Simon: Methods of modern mathematical Physics vol 1: Functional analysis. (Lots of racy slick proofs in here.)

Angus Taylor: Introduction to Functional analysis. (Can be a bit slow but it’s got all the details.)

Covid 19 will make the mechanics of the course a bit non-standard. The individual lectures, or some approximations thereof, will be posted HERE as the course proceeds. They will also be updated from time to time. It will be very helpful to you if you can look at the lecture on line before it is given. Even a superficial reading is useful.

• Generalities
• Index
• Hahn Banach take 1
• Normed vector spaces
• Holder Minkowski
• Holder Minkowski lecture
• Continuous maps and duals
• Hilbert space

Look for homework assignments HERE

• Homework1
• Homework2
• Potential projects

Midterm, grades

So that I have some idea how much students are understanding we will have a midterm in class some time in September if I can figure out how to do it…..

Otherwise grades will be assigned on the basis of a short (2-3 pages) essay on a topic not necessarily covered in the course.