Functional Analysis (602, Real Analysis II), Fall 2010

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.

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.

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