### Math 516 - Linear Analysis

### Instructor

### Office hours

### Syllabus

This core course is an introduction to the basics of functional analysis:
- normed spaces and Banach spaces;
- the principles of functional analysis: the Hahn-Banach theorem, the uniform boundedness principle, the open mapping
theorem, the closed graph theorem;
- Hilbert spaces, and the spectral theorem for compact, self-adjoint operators on Hilbert space;
- fixed point theorems (Banach, Schauder) with applications, e.g. to differential and integral equations.

I will put a lot of emphasis on examples and applications that demonstrate the elegance and flexibility
of functional analytic methods: how the Hahn-Banach theorem can be used to prove approximation theorems, how the Baire category
theorem almost effortlessly yields continuous functions that are nowhere differentiable, and how fixed point theorems can be used
to solve differential and integral equations.

Time permitting, I will also cover topics not included in the 516 syllabus such as locally convex vectors paces, weak topologies, and
weak* topologies.

### Textbooks

None required, but

J. B. Conway, *A Course in Functional Analysis*. Springer Verlag, 1985.

is recommended. Other books that can be used as supplementary texts are

- B. Bollobás,
*Linear Analysis. An Introductory Course*, Second Edition. Cambridge University Press, 1999.
- N. Dunford and J. T. Schwartz,
*Linear Operators*, I. Wiley-Interscience, 1988.
- G. K. Pedersen,
*Analysis Now*. Springer Verlag, 1989.
- W. Rudin,
*Functional Analysis*, Second Edition. McGraw-Hill, 1991.

For the background in linear algebra

P. R. Halmos, *Finite-dimensional Vector Spaces*. Springer Verlag, 1974.

is an excellent source.

As the course progresses, I also plan to produce TeXed lecture notes.

### Grading

The grade will be based on weekly homework assignments (30%), an in-class-midterm on October 18 (20%), and a final (50%).

Last update: 9/05/02.