**Instructor:**
Roman Vershynin

Office: 4844 East Hall

E-mail: romanv "at" umich "dot" edu

**Class meets:** MWF 1:10-2:00 in 3866 East Hall.

**Office Hours:** MW 2:10-3:00 in 4844 East Hall.

**Prerequisites:**
Intro to Topology (Math 590)
and Analysis II (Math 597).

**Lecture Notes:**
Lecture Notes in Functional Analysis. The course will follow these notes.

**Course Contents:**

- I. Banach spaces
- Linear spaces. Subspaces and quotient spaces. Linear operators
- Normed spaces. Examples: ell_infinity, c_0, c, ell_1, C(K), L_1
- Convexity of norms and balls. Minkowski inequality. Spaces L_p, ell_p
- Subspaces and quotient spaces of normed spaces. L_infinity
- Banach spaces. Completeness criterion via series; completeness of L_p
- Fixed points of contractions
- II. Hilbert spaces
- Inner products. Cauchy-Schwartz inequality. Spaces L_2, ell_2
- Orthogonal projections and orthogonal decompositions
- Orthonormal bases. Fourier series. Bessel's inequality and Parseval's identity
- Gram-Schmidt orthogonalization. Isomorphism of all Hilbert spaces
- III. Bounded linear functionals and operators
- Bounded linear functionals. Norm. Dual space
- Riesz representation theorem. Application: von Neumann's proof of Radon-Nikodym theorem
- General form of linear functionals on classical spaces
- Extensions of linear functionals. Hahn-Banach Theorem. Second dual space. Separation of convex sets
- Applications of Hahn-Banach theorem: Banach limit, invariant means
- Bounded linear operators. Norm. Space of bounded linear operators. Isomorphisms and isometries
- Extensions of linear operators. Projections
- Adjoint operators.
- IV. Main principles of Functional Analysis
- Open mapping theorem. Isomorphisms and equivalent norms
- Finite dimensional Banach spaces
- Closed graph theorem
- Banach-Steinhaus theorem. Applications in Fourier analysis
- Compact sets in Banach spaces. Compactness criteria in concrete spaces
- Weak and weak* convergence and topologies.
- Alaoglu theorem
- Krein-Milman theorem
- V. Elements of spectral theory
- Compact operators
- Spectrum: definition and properties
- Spectrum of compact operators
- Selfadjoint operators in Hilbert space
- The spectral theorem for compact selfadjoint operators
- Hilbert-Schmidt operators
- Functional calculus of self-adjoint operators.
- Unitary operators. Polar decomposition
- Spectral theorem for bounded self-adjoint operators.

**Recommended texts:**
Some of the course material will be taken from these two books, especially [EMT].
Both books are not required.
Most of the time the course will not follow any existing textbook.
I am planning to scan and post my own notes after each lecture (scroll down).
But they may be too sketchy as I am writing them for myself. So, please take notes.

- [RS] M. Reed, B. Simon, Methods of modern mathematical physics. I. Functional analysis. Second edition. Academic Press, Inc., New York, 1980. ISBN: 0-12-585050-6
- [EMT] Yu. Eidelman, V. Milman, A. Tsolomitis, Functional analysis. An introduction. Graduate Studies in Mathematics, 66. American Mathematical Society, Providence, RI, 2004. ISBN: 0-8218-3646-3

**Assignments:**
**Homework** will be assigned every Friday (scroll down).
Some of the problems will repeat or resemble those from the book [EMT].
Try to work on them without looking at their solutions in [EMT] first.
The course grade will be based on three **take-home exams** (scroll down).
You must work individually on all exam problems.

**Course webpage:**
http://www-personal.umich.edu/~romanv/teaching/2010-11/602/602.html