Note: This syllabus was revised by Steve Shkoller and approved by the GPC at its meeting dated October 15, 2007. It was posted on the website on November 13, 2007.

Course Request Summary

Department Submitting Request: Mathematics

Request for Action: CHANGE

Effective: 200210

Course Subject Area: Mathematics

Subject Code: MAT

Course Number: 201C

Descriptive Title: Analysis

Abbreviated Title: Analysis

Units: 4

Learning Activity

1st LEC 3.0 hrs/wk

2nd T-D 1.0 hrs/wk

In Progress Grading: None

Consent of Instructor: No

Prerequisite(s): Graduate standing or consent of instructor

Restrictions on Enrollment:

Course Description: Metric and normed spaces. Continuous functions. Topological, Hilbert, and Banach spaces. Fourier series. Spectrum of bounded and compact linear operators. Linear differential operators and Green’s functions. Distributions. Fourier transform. Measure theory. L^p and Sobolev spaces. Differential calculus and variational methods.

General Education: No GE certification.

Topical Breadth:

Diversity:

Writing Experience:

Cross Listing:

Justification:

Repeat Credit:

Credit Limitations:

Mode of Grading: Letter

Quarters to be Offered: III

Instructors Name(s): Staff, Chair in Charge

Title(s):

Remarks:

Expanded Course Description

TOPICAL OUTLINE:

1. L^p spaces

- Basic inequalities of Jensen, Holder, Minkowski

- Completeness

- (Toy version of Radon-Nikodym for 1 lecture)

- Continuous linear functionals and weak convergence

- Lower semi-continuity of norms and uniform boundedness (weak convergence)

- Dual space of L^p

- Convolution on L^p

2. Integral inequalities

- Young’s inequality and the Hardy-Littlewood-Sobolev inequality for L^p functions

3. Fourier Transform on functions

- L^1 Fourier transform

- Plancherel’s Theorem

- L^2 Fourier transform

- Inversion formula

- Hausdorff-Young inequality

4. Distributions

- The space of test functions (either C^\infty_0 or Schwartz class functions)

- The space of distributions or tempered distributions

- Convergence in distributional sense

- Locally summable functions in L^p (and in particular L^1)

- Distributional derivatives and weak derivatives

- Density of smooth function in the space of distributions

- The Sobolev space W^{1,p} and its dual space

5. The Sobolev Spaces H^k

- Definitions, completeness and local/global approximation theorems

- Weak derivatives

- Integration by parts formulas

- Trace theorem and H^{1/2} (optional)

- Weak Convergence

- Sobolev inequalities (proofs if time permits)

READING:

Analysis by Lieb and Loss, Chapters 2, 4-8 and Applied Analysis by Hunter and Nachtergaele, Chapters 10-12

GRADING PERCENTAGES:

40% in-depth problem assignments, 20% midterm, 40% final exam.

COURSE FORMAT AND REQUIREMENTS:

This course meets for 10 weeks.

EXPLANATION OF POTENTIAL COURSE OVERLAP:

None.

GENERAL EDUCATION JUSTIFICATION:

None.

ADDITIONAL INFORMATION FOR STUDENTS:

The goal of the 201ABC sequence is to provide graduate students in mathematics, applied mathematics, and related areas with a firm foundation in the principles of analysis. The sequence covers the most common and important techniques of analysis used in other branches of mathematics as well as in a wide variety of applications in science and engineering.

In 201C, Green’s functions and the spectral theory for compact self-adjoint operators are used to study Sturm-Liouville operators, and important class of unbounded self-adjoint operators. The theory of distributions and the Fourier Transform, a beautiful subject with an astounding array of applications, is studied in detail. There is a short introduction to measure theory (a complete treatment is given in a separate course, MAT 206), leading to the study of L^p spaces. The last topic is differential calculus in Banach spaces and the calculus of variations.