## Course on Functional Analysis Vanderbilt Fall 2020

Functional Analysise.

### 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.

Covid 19 will make the mechanics of the course a bit non-standard.
The individual lectures will be posted HERE as the course proceeds.
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.

Course Notes 1

** Look for homework assignments HERE**

Potential projects

### Textbook:

The text for the course will be Lax's "Functional
Analysis"
### Midterm, grades

So that I have some idea how much students are understanding we
will have a midterm in class some time in September.
Otherwise grades will be assigned on the basis of a short (2-3 pages) essay on a topic not necessarily covered in the course.