The third edition of this book hasn't been published yet (so you can't buy a copy at the bookstore or anywhere else). We will be testing a preliminary version, which you can get from the Math 321 site on Blackboard.
Please read the note below about reading this textbook.
Abstract mathematical reasoning in the context of analysis in Euclidean space. Introduction to formal reasoning, sets and functions, and the number systems. Sequences and series; Cauchy sequences and convergence. Required for all mathematics majors. Prereq: MATH 223.
Math 321 will cover roughly the first five chapters of the book, skipping a few sections and possibly adding some additional material not in the book. Math 322 will pick up where 321 leaves off and cover much of the rest of the book.
Note: In Math 321 we will cover and use some basic ideas from linear algebra (the definitions of a real vector space and inner product space); Math 322 will use some slightly more substantial ideas (the matrix of a linear map). You might notice that in the preface, the textbook authors say that "course material for this book is normally taken by students . . . after they have completed . . . a course in linear algebra." I, however, don't assume that you've already taken a linear algebra course. We will cover everything about linear algebra that's needed in the class. Of course students who have already taken Math 307 will have the advantage of seeing those ideas for the second time.
Here's a rough, tentative schedule for the class:
| Topics | Book sections | Weeks |
|---|---|---|
| Sets and functions | Introduction | 1 |
| The real line, convergence | 1.1-1.5 | 1-3 |
| Euclidean spaces | 1.6-1.7 | 4 |
| Open and closed sets | 2.1-2.6 | 4-5 |
| Sequences and series | 2.7-2.9 | 6-7 |
| Compactness and connectedness | 3.1-3.5 | 7-9 |
| Continuity | 4.1-4.8 | 10-12 |
| Uniform convergence | 5.1-5.4, 5.8 | 13-14 |
| Additional topics (as time allows) | 5.5, 5.10, 5.7, other topics not in the book | 14-15 |
Students may work together on homework. However, each student must figure out how to write up his or her own solution to be turned in. That means for example that you and a friend may figure out together how to prove a statement, but the written-out proofs you turn in should not be verbatim copies of each other.
All homework assignments are due in class on the listed date. Late homework will not be accepted. If unusual circumstances arise, we can discuss alternative arrangements if you contact me in a timely manner.
A typical textbook for a math course at this level follows a scheme along the lines of definition-discussion-theorem-proof, possibly repeated several times in each section. This book, on the other hand, has sections which are organized like definition-discussion-theorem-more discussion, and all the proofs of the theorems are collected in a separate section at the end of each chapter. Don't make the mistake of thinking this means you don't need to read the proofs! One of the main goals of this class is for you to learn to understand and write proofs like the ones in this book. In particular, your homework solutions should consist mostly of proofs like those at the ends of the chapters.
Instead, you should first read a section of the book, which is written aiming to help you learn how to think about the definitions and theorems (the "trade secrets", as the authors say in the preface); then after that you should tackle the proofs of the theorems, where you will see how to work with the definitions and earlier theorems rigorously. Due to limited time in lectures I will sometimes emphasize either proofs or discussion of concepts and spend less time on the other, but you are responsible for learning about both.
Each chapter of the book also contains a section of additional "worked examples", essentially exercises with complete solutions, right before the theorem proofs. You can read many of these before the theorem proofs, but some of them build on ideas in the proofs. I recommend looking at them when you're beginning to work the homework. (The chapter-by-chapter organization means it's a bit of work for you to identify which examples are related to the current section, but doing that work is also good for you.)