Note: Course was revised by Steve Shkoller and approved by the GPC on October 15, 2007.

 

 

Course Request Summary

 

Department Submitting Request: Mathematics

Request for Action: CHANGE

Effective: 200210

Course Subject Area: Mathematics

Subject Code: MAT

Course Number: 201A

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: I

Instructors Name(s): Staff, Chair in Charge

Title(s):

 

Remarks:

 

 

 

 

Expanded Course Description

 

TOPICAL OUTLINE:

  1. Metric and normed spaces (review)

-         Metrics, norms, limits, liminf, limsup

-         Pointwise, uniform, and norm convergence

-         Continuity and completeness

-         Compactness in finite-dimensions

-         Compact and Locally Compact Spaces

  1. Spaces of continuous functions

-         Definitions of spaces

-         Convergence

-         Tychonoff’s Theorem

-         Arzela-Ascoli Theorem

-         Stone-Weierstrauss Theorem

  1. Topological Spaces

-         Definition of topological spaces

-         Bases of opens sets

-         Comparing topologies

  1. Banach Spaces

-         Normed vector spaces

-         Linear functionals and bounded linear maps

-         The kernel and range of linear maps

-         Convergence in the space of bounded linear operators

-         Dual spaces

  1. Hilbert Spaces

-         Inner products

-         Orthogonality and projections

-         Orthonormal bases

-         Applications

 

READING:

Applied Analysis by Hunter and Nachtergaele, Chapters 1, 3-6

 

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.

 

201A lays the foundations for the analysis of function spaces and maps between such spaces. The notions of metric spaces, normed spaces, and general topological spaces are introduced and their basic properties are explored. A variety of applications in mathematics and applied mathematics are discussed in detail.