**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).

**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.

**Lecture notes:**

- Sep 9. Prerequisites, sources used. Linear spaces. Origins of functional analysis.
- Sep 11. Quotient spaces. Linear operators. Normed spaces.
**Homework.** - Sep 14. Convexity of norms and balls. L_p spaces. Minkowski inequality.
- Sep 16. Subspaces and quotient spaces of normed spaces. The space L_infty.
- Sep 18. Banach spaces. Criterion of completeness in terms of series.
**Homework.** - Sep 21. Hilbert spaces. Cauchy-Schwartz inequality.
- Sep 23. The spaces L_2, ell_2. Orthogonal projections.
- Sep 25. Orthogonal systems and orhtogonal series.
**Homework.** - Sep 28. Fourier series. Bessel's inequality, Parseval's identity.
- Sep 30. Gram-Schmidt orthogonalization. Isometry between Hilbert spaces.
- Oct 2. Bounded linear functionals. Dual space.
**Exam 1.** - Oct 5. Riesz representation theorem. Von Neumann's proof of Radon-Nikodym theorem.
- Oct 7. Bouned linear functionals on classical spaces. Extensions by continuity.
- Oct 9. Hahn-Banach theorem. Supporting functionals.
**Homework.** - Oct 12. Second dual. Separation of convex sets.
- Oct 14. Bounded linear operators.
- Oct 16. Extensions and projections. Adjoint operators.
**Homework.** - Oct 21. Open mapping theorem.
- Oct 23. Inverse operators. Finite dimensional Banach spaces.
**Homework.** - Oct 26. Closed graph theorem.
- Oct 28. The principle of uniform boundedness.
- Oct 30. Compact sets in Banach spaces.
**Exam 2.** - Nov 2. Weak convergence.
- Nov 4. Weak topology. Weak* convergence and topology. Alaoglu's theorem.
- Nov 6. Krein-Milman theorem.
**Homework.** - Nov 9. Compact operators.
- Nov 11. Fredholm alternative.
- Nov 13. Classification of spectrum.
**Homework.** - Nov 16. Properties of spectrum and resolvent.
- Nov 18. Spectral radius. Spectrum of compact operators.
- Nov 20. Self-adjoint operators. Quadratic forms.
**Homework.** - Nov 23. Spectrum of self-adjoint operators. Spectral theorem for compact selfadjoint operators.
- Nov 25. Hilbert-Schmidt operators.
- Nov 30. Order and functional calculus for self-adjoint operators.
**Exam 3.** - Dec 2. Polar decomposition. Unitary operators.
- Dec 4. Orthogonal projections. Resolutions of identity.
- Dec 9. Spectral projections.
- Dec 11. Spectral integral. The spectral theorem for bounded self-adjoint operators.

**Course webpage:**
http://www-personal.umich.edu/~romanv/teaching/2009-10/602/602.html