# MATH 621 Introduction to Analysis I

### Section 001, Fall 2017

 Instructor: Kevin McLeod Office: EMS E481 Office hours: MW 2:00-3:15 PM;                      TR 10:00-11:00 AM                      (or by appointment) Phone: 229-5269 E-mail: kevinm@uwm.edu Home page: sites.uwm.edu/kevinm
Class schedule: TR 11:00-12:15, NWQ 1961

Text: Walter Rudin, Introduction to Analysis (3rd Edition).

### Course Description

The 2-semester sequence MATH 621/622 is an Introduction to Analysis, but what is the meaning of “analysis” in this context? Oversimplifying considerably, analysis is the part of mathematics which uses limits in an essential manner—as opposed to algebra, in which limits rarely appear. You probably first encountered the mathematical concept of “limit” when you studied calculus, and one way of thinking about analysis at the level we will study it this year is that it is calculus “done right”: we will see plenty of material that is new to you, but we will also take care to fill in any gaps or omissions in proofs of results that you are already familiar with from your earlier sequence of calculus courses.

Analysis is an essential part of any working mathematician’s toolkit, as well as being a beautiful subject in its own right. Unfortunately, it can also be a subject in which the technical details are allowed to obscure the beauty and simplicity of the fundamental ideas. The way to avoid this is to work through and discuss lots of examples and problems, which is why you will be expected to participate in class discussions, and perhaps even lead some of those discussions yourself.

The core of MATH 621 will consist of (most of) the first 6 chapters of the text. Beginning with some foundational work on the real number system, these chapters cover continuity, differentation, and integration of functions of one variable. (Sound familiar?). I hope to comnplete the rest of the text in MATH 622, but it is more important that we cover material well than that we finish the book, so you should let me know if we are moving too quickly. In particular, the final chapter, Chapter 11, is on the Lebesgue theory of integration and is definitely optional. (That material is covered in great depth in MATH 711.)

I will try to keep lecturing to a minimum, and devote class time to discussion and problem solving. As a result, you will be expected to read the textbook in a timely fashion, as necessary to participate in the class discussions. When I assign reading for a class, I will expect you to e-mail me, at least 24 hours prior to the class, with one or more questions you have about the reading. I will review those e-mails before the class and prepare a brief lecture or other activity for any common questions, as I feel necessary. I will also save those e-mails; they will be used at the end of the semester as evidence that you were completing the reading assignments, and will count towards the homework portion of your grade.

Almost all class information will be posted on the class website, https://sites.uwm.edu/kevinm/math-621-introduction-to-analysis-i/. All class information (homework, class cancellations, etc.) will be posted on the website; some will be posted only there. If I find useful and relevant links during the semester, I will post them as well; if you find some yourself, please let me know. You are responsible for any information posted on the website, so please check it frequently.

### Evaluation

Your grade for the course will be based on the following factors:

• Homework In addition to the textbook (and possibly other) readings, you will be assigned written homework regularly, some of which you will be expected to hand in. Of the homework you hand in, approximately half will be treated summatively; i.e. you will be given a grade which will contribute to your final grade for the course. (The other half will be treated formatively: you will be given feedback, but no score.) 25%.
• Class participation You will be expected to contribute to the class discussion, to the extent of leading discussions on topics from the text or homework problems. (For this reason alone, regular class attendance will be essential.) 25%.
• Exams There will be two exams, a midterm and the final exam. (Please note the scheduled final exam time for this class: 10:00-12:00 noon on Friday, December 22.) 25% each.

### Average Time Investment

The amount of time that an average student should expect to spend on this class is as follows:

• Classroom time (face to face instruction): 45 hours
• Time taking exams (midterm, final exam): 15 hours
• Time completing reading and other homework assignments: 80 hours
• Time for preparation and study for exams: 10 hours

Total number of hours: 150.

### Students with disabilities

If you feel you are a student with a disability, please feel free to contact me early in the semester for any help or accommodation you may need.

### University policies

The Secretary of the University has a page dedicated to university policies for religious observances, grade appeal procedures, military service and other matters. You should also familiarize yourself with the information on the Dean of Students Office webpage concerning proper student conduct at the university, both academic misconduct and non-academic misconduct. You will be held responsible for the information and policies contained at these links.