# MATH W4061 (Fall 2006)

INTRODUCTION TO MODERN ANALYSIS

General Course Information:

MW 09:10am - 10:25am

Mathematics Hall 312

Instructor: H. Pinkham

## Prerequisites

Calculus IV (MATH V1202) or the equivalent (e.g. Honors Mathematics III and IV, MATH V1207- MATH V1208) and Linear Algebra (MATH V2010) - waived for students having taken Honors Math III and IV.

## 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 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, and dropped off in a box across from 409 Mathematics. There will be weekly recitations run by the teaching fellows in the course, where problems similar to the assigned problems will be worked out.

## Material Covered

Real numbers; metric spaces and elements of general topology; numerical sequences and series; continuity and differentiation; the Riemann Stieltjes integral; uniform convergence.

## Method of Evaluation

In-class midterm on Wednesday October 18, worth 35% of the total grade, covering the material of sections I and II (the first two chapters of Rudin). Final exam, worth 45% of the total grade, on the material of sections III through VII. Note that the final is not cumulative.

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 Text

Walter Rudin, Principles of Mathematical Analysis (McGraw-Hill). Simply called “Rudin” in these notes. We will cover chapters 1 through 6, and the beginning of chapter 7. We will also do the examples from the beginning of chapter 8. We will skip the small amount of material on complex numbers. Rudin is a great reference book, but it is dry, concise and difficult for beginners.

## Syllabus

Outline of the class meetings:

## Section I: The Real Number Field | ||

1. Sep-6-06 | Primarily review. Notation, sets and operations on sets (union, intersection, complements), proof techniques, functions. | |

2. Sep-11-06 | Set theory. Ordered sets. Greatest lower bound (glb) and least upper bound (lub) Rudin p.3-5 | |

3. Sep-13-06 | Fields. Ordered fields, the archimedean property, the real field. Rudin p. 5-12 | |

4. Sep-18-06 | Euclidean space. The Schwarz inequality and the triangle inequality. Rudin, p. 16-17 plus the Schwarz inequality (in the real case only) p. 15-16 | |

## Section II: Metric Spaces and Topology | ||

5. Sep-20-06 | Cardinality of a set. Countable and uncountable sets. Rudin p. 24-30 | |

6. Sep-25-06 | Metric spaces. The basic definitions, open and closed sets. Rudin p. 30-36 | |

7. Sep-27-06 | Compact sets. Rudin p. 36-40 | |

8. Oct-2-06 | More on compact sets: the Heine-Borel theorem. Rudin p. 36-40 | |

9. Oct-4-06 | Connected sets and the Cantor set. Rudin p.41-43 | |

## Section III: Numerical Sequences and Series | ||

10. Oct-9-06 | Convergent sequences and subsequences. Cauchy sequences and completeness. Examples. Rudin p. 47-58 | |

11. Oct-11-06 | The real number system: the construction of the real field. The appendix of chapter 1 of Rudin. | |

12. Oct-16-06 | Series. Rudin p. 58-65 | |

13. Oct-18-06 | Midterm on the material of sections I and II. | |

14. Oct-23-06 | Convergence tests. Power series. Rudin p. 65-78 | |

## Section IV: Continuity | ||

15. Oct-25-06 | Continuity in a metric space. Rudin p. 83-89 | |

16. Oct-30-06 | Continuity and compactness. Uniform continuity. Rudin p. 89-93 | |

17. Nov-1-06 | Continuity and connectedness. Monotone functions. Rudin p. 93-98 | |

## Section V: Differentiation | ||

18. Nov-8-06 | Derivatives. Rudin p. 103-106 | |

19. Nov-13-06 | Mean value theorem. Rudin p. 107-113 | |

## Section VI: Integration | ||

20. Nov-15-06 | Definition of the integral. including the Riemann-Stieltjes integral. Rudin p. 120-124 | |

21. Nov-20-06 | What functions are Riemann integrable. Rudin p. 124-127 | |

22. Nov-22-06 | Properties of the integral. Rudin p. 128-133 | |

23. Nov-27-06 | Integration and differentiation. The Fundamental Theorem of Calculus. Rudin p. 133-7 | |

## Section VII: Sequences and Series of Functions | ||

24. Nov-29-06 | Sequences and series of functions. Rudin p. 143-147 | |

25. Dec-4-06 | Uniform convergence and continuity of functions. Rudin p.147-151 | |

26. Dec-6-06 | Continuation of uniform convergence and continuity of functions. Rudin p. 147-151 | |

27. Dec-11-06 | Last day of class. Review. |