Math 8601 Real Analysis
2007-08

Location and time: Vincent Hall 209, MWF 12:20pm-1:10pm

Text: Real and Functional Analysis, 3rd edition, GTM 132, Springer, by Serge Lang.

Reference books:

Lecturer: Tian-Jun Li, Vincent Hall 260, (612)625-2036
Email: tjli@math.umn.edu URL: http://www.math.umn.edu/~tjli
Office hours: 11-12, Thursday.

Grader: Lu Li, Vincent Hall 454, (612)624-3531
Email: lixxx415@math.umn.edu
Course Content

In this year long course I plan to cover Ch. 1-3, 4-8, 10, 13-15, 17, 21-23 of Lang's book. Beyond the book I will cover basics of Sobolev spaces.

In the fall semester the emphasis is on integration and Functional analysis, I aim to cover the following topics:
Ch. 1-3 General Topology
Ch. 4 Banach spaces (Hahn-Banach Theorem , Banach algebra, completion)
Ch. 5 Hilbert spaces (orthonormal basis, orthogonal projection, Rieze representation, weak convergence)
Ch. 6 General integral (abstract measures, measurable maps, positive measures on algebras and their extensions to \sigma-algebras, fundamental lemma of integration, completion of step maps, basic convergence theorems, approximations, Fubini Theorem, Lebesgue integral)
Ch. 7 Duality and representation (L^2 space, duality between L^1 and L^{\infty}, L^p spaces with 1< p< \infty)
Ch. 8 Applications of integration (convolution, continuity and differentiation under the integral sign, Dirac sequences, Schwartz space and Fourier transform)
Ch. 11 Distribution
Ch. 15 Open mapping theorem and its applications (Baire category Theorem, open mapping theorem, closed graph theorem, principle of resonance, topological complement)
Ch. 17 Compact and Fredholm operators (compact operators, Fredholm operators and the index, stability of Fredholm operators, spectral theorem for compact operators, applications to integral equations)
Homework

Homework will be assigned in class, and be due on Monday of the following week.
You are permitted and even encouraged to discuss homework problems with classmates.

      
Homework assignments

Tests
There are three in class exams. The first is on Oct. 5. The second one is on Nov. 12.

      
(Partial) Solutions to Exam 1

Grading
Grades will be based on Homework (40%) and 3 exams (20% each).
Feedback and Questions
You are very welcome to visit me during my office hours. You can also make appointments to see me at other time.