# MATH W4062 (Spring 2007) INTRODUCTION TO MODERN ANALYSIS

General Course Information:
MW 09:10am - 10:25am
Mathematics Hall 312
Instructor: H. Pinkham

## Prerequisites

This is the second semester of a year-long course. You must have taken the first semester of this course (Math 4061) this year, or the equivalent. Students who did not take Math 4061 this year must get in touch with the instructor as soon as possible.

To paraphrase the introduction of The Cauchy-Schwarz Master Class by J. Michael Steele (Cambridge University Press), the most important prerequisite for benefiting from this course is the desire to master the craft of discovery and proof.

## Course Objectives

This is a proof-based course. It is central to a good understanding of mathematics, and plays a role parallel to that of the Introduction to Modern Algebra, W4041-2, with the additional twist that it is even more important for the applications of mathematics . It is essential to master the material of this course for graduate school in mathematics of course, but also for graduate study in many other fields, including economics.

My expectation is that you will learn the definitions and the statements of the theorems. I will also expect you to be able to prove the principal theorems, and work examples and counterexamples. This can only be done by reading the book very carefully, attending the classes and doing (that is, struggling with) the homework.

Homework problems will be distributed every Wednesday in class, and due at 5 PM on Wednesday a week later. No late homework will be accepted. Homework can either be turned in in class on Wednesday morning, or dropped off in a box across from 409 Mathematics.

## Material Covered

1. Review of complex numbers, power series, the classic functions of analysis, Fourier series. Rudin chapter 8, omitting the section on the Gamma function. For Fourier series, Carothers chapter 15.

2. The metric space C(X) of continuous functions on a compact metric space X. Rudin chapter 7 and Carothers chapters 10-12.

3. Differentiation in several variables, with emphasis on the inverse and the implicit function theorems. Rudin chapter 9.

4. The Lebesgue integral: Rudin chapter 11 and Carothers chapters 16-18.

As time permits: more on convexity; a discussion of other function spaces.

## Method of Evaluation

In-class midterm on Monday March 5th, worth 35% of the total grade. Final exam, worth 45% of the total grade, on the material covered after the midterm. Weekly homework: 10% of the grade. Note that no late homework will be accepted. Two unannounced in-class quizzes, each worth 5% of the grade.

## Required Texts

Walter Rudin, Principles of Mathematical Analysis (McGraw-Hill). Simply called “Rudin” on this site. Last semester we covered the first 6 chapters, plus the first half of chapter 7. This semester we will cover the rest of the book, excluding chapter 10, which we will not cover at all.

As you know, Rudin is a great reference book, but it is dry, concise and difficult for beginners. For that reason I am asking you to get a second book: N. L. Carothers, Real Analysis (Cambridge University Press). This book is easier to read and gives additional motivation and examples. It also gives a history of the development of the field. I will be following Carother’s presentation for the metric space C(X) of continuous functions on a metric space X, and the Lebesgue integral.

## Syllabus

Outline of the class meetings:

 Jan-17-07 Complex numbers. R p.12-16. Jan-22-07 Algebraic closure of the Complex Field. R, p.184-185 Uniform convergence and integration. R, p.151-2 ; Uniform convergence and differentiation. R, p.152-3. Jan-24-07 Power series. R, p. 172-178 Jan-29-07 exp, ln, trig functions. R, p. 178-184 Jan-31-07 The space C(X) of complex-valued, continuous and bounded functions with domain a metric space X. Reread R, 7.14 and 7.15, p.150-151. Read chapter 10 of Carothers, which covers the same material as the first half of chapter 7 of R. Feb-5-07 Equicontinuity and its connection to C(X). R. p. 154-158. C, p.178-182. Feb-7-07 The Arzelà-Ascola Theorem Feb-12-07 The Weierstrass approximation theorem. R. 159-160. C, 162-168. In class I will give the Bernstein proof of the theorem, namely, the one in C. Feb-14-07 The Stone-Weierstrass theorem. R, p. 161-165, C. p.188-201 Feb-19-07 Functions of bounded variation. C chapter 13, p. 202-212 Feb-21-07 Fourier series, I. R, p.185-192 (note that we will not cover the section on the Gamma function). C, p.170-176 (Weierstrass’s second approximation theorem) and C chapter 15. Feb-26-07 Fourier series, II Feb-28-07 Fourier series, III. Mar-5-07 Midterm. Mar-7-07 Fourier series, IV Mar-19-07 Functions of several variables, I. R, chapter 9. You need to review linear algebra and multivariable calculus before attempting the reading. Rudin, pp. 204-211. Mar-21-07 Differentiation in several variables: partial derivatives. Rudin, pp. 211-220. Mar-26-07 Functions of several variables, III : the inverse function theorem. You need to review the contraction principle before reading this. Rudin, pp. Mar-28-07 Functions of several variables, III : the implicit function theorem. Remember that there will be a quiz at the beginning of class. Apr-2-07 Functions of several variables, IV : conclusion Apr-4-07 Lebesgue measure, I. C, chapter 16. R, chapter 11 is an alternate reference for the material in the rest of the course, but I will follow the presentation in C. Apr-9-07 Lebesgue measure and measurable functions, II. Apr-11-07 Measurable functions, II C, chapter 17 Apr-16-07 Lebesgue integral, I C, chapter 18 Apr-18-07 Lebesgue integral, I C, chapter 18 Apr-23-07 Lebesgue integral, II Apr-25-07 Lebesgue integral, III Apr-30-07 Lebesgue’s differentiation theorem. C, p. 359-376.