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: 201B

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. Lp and Sobolev spaces. Differential calculus and variational methods.

General Education:

Topical Breadth:

Diversity:

Writing Experience:

Cross Listing:

Justification:

Repeat Credit:

 

Credit Limitations:

Mode of Grading: Letter

Quarters to be Offered: II

Instructors Name(s): Staff, Chair in Charge

Title(s):

 

Remarks:

 

 

 

 

Expanded Course Description

 

TOPICAL OUTLINE:

  1. Basic measure and integration theory

-         Fundamental definitions from measure theory (proofs left to 206)

-         Measurable functions and approximation by simple functions

-         Dominated and Monotone Convergence Theorems and Fatou’s Lemma

-         Fubini and Tonelli Theorems

-         Definition of L^p and l^p spaces and concrete examples of L^2 Hilbert spaces

  1. Fourier Series

-         Definitions and properties

-         Sobolev spaces H^s of periodic functions on torus for s real

-         Poisson summation/integral formula for the disk and the Dirichlet problem

  1. Bounded linear operators on Hilbert spaces

-         Othogonal projections

-         Dual space of Hilbert space and representation theorems

-         Weak convergence in Hilbert space, Banach-Alaoglu Theorem

  1. Spectrum of Bounded Linear operators

-         Diagonalization of matrices

-         Spectral theorem for compact, self-adjoint operators

-         Compact operators

-         Fredholm Alternative Theorem

-         Functions of operators

  1. Calculus on Banach Spaces

-         Bochner integrals

-         Derivatives of maps on Banach spaces

-         The calculus of variations

 

READING:

Analysis by Lieb and Loss, Chapter 1, and Applied Analysis by Hunter and Nachtergaele, Chapters 7-9 and 13

 

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.

 

201B is devoted to Hilbert spaces. Topics include orthonormal bases and associated expansions such as Fourier series, and the spectral theory of linear operators on Hilbert spaces.

 

ENTRY LEVEL:

Graduate standing or consent of instructor.