Functional Analysis (602, Real Analysis II), Fall 2009

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.

Additional texts used in preparation of this course are listed in Lecture 1 below.

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:

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

Course webpage: