Math 331: Foundations of Analysis

About Math 331: Goals

This course is an introduction to basic analysis of functions of a single variable. It presents a careful development of the real number system and the theory of calculus on the real line. This can be followed by further study in subsequent courses with extensions of the theory to Rⁿ and the complex plane. The course is designed to be a rigorous encounter with formal mathematics for serious mathematics students who have taken a year of calculus (and preferably a third term of calculus) and additional course work in abstract mathematics including an introduction to proofs and linear algebra.

Early mathematical education up through calculus focuses on learning how to carry out various mathematical operations. But the heart of mathematics does not lie in reproducing computational procedures or memorizing mathematical facts but in seeing why the facts are true, in grasping the arguments which show why the computation works, in understanding the connections which weave the individual pieces of mathematics into a theory. In Math 135, you were introduced to the proof techniques that are used to construct these arguments.

The discovery of this "heart" of mathematics begins by cultivating a deep curiosity about what underlies the methods and conclusions of mathematics. A desire to investigate how calculus is put together and what makes it tick is the primary prerequisite, along with mathematical maturity, for this course.

In developing this course with Professor Belding, we had three major goals in mind.

First, we wished to help students develop some understanding and control of the language of mathematics: definitions, theorems, and proofs.
Secondly, we wanted to present the central concepts and theory of calculus thoroughly and clearly enough to provide a secure foundation for more advanced mathematics courses and enable students to grasp some of the unity and beauty of the subject.
Finally, we wished to foster an appreciation for the living, human nature of mathematics, and a sense of how mathematics is created and evolves.

The transition to formal mathematics based on rigorous proofs is a necessary but difficult step. Reading and writing proofs requires a new level of precision and attention to detail. Rigorous definitions force us to rework our vague intuitive understanding of concepts such as limits or integrals.

This course will be an intellectual challenge and you should be proud of your achievement at the end of the term. You should feel that you could construct "the Calculus" starting from a small set of basic principles. You should feel ready to extend the concepts that you encounter to new situations and that your toolbox of mathematical proof techniques has been greatly enhanced. In short, you should feel like you are becoming a mathematician.

Text

Professor Belding and I wrote the text for this course with the students at Hobart and William Smith in mind. In this text we try to provide the demanding but supportive environment needed to make the transition to formal mathematics based on rigorous proofs successful.

We supply a context for the more difficult definitions and proofs by tracing the history behind the concepts. Hidden problems which motivate the design of a proof or wording of a definition are pointed out and discussed. The proofs have been crafted with the student in mind. Thus, many details are included which draw attention to points that may seem obvious or repetitive to more veteran mathematicians.

There are lots of problems! Nearly all problems require a proof of some kind. Some are simply intended to guide your review of material. These often require only slight modification of earlier arguments or immediate reflection on the meaning of a definition or theorem. However, written answers to such straightforward questions serve to reveal serious misconceptions early on. When an involved argument is required, the problem is frequently broken into steps to point toward fruitful questions and to reveal the structure and level of detail needed for a thorough proof.

This text does not pretend to be a comprehensive treatment of basic analysis. Instead, we have tried to present the main current of analysis faithfully and with enough spirit to inspire you to explore further.

Prerequisites and Expectations

This course assumes that you are (or are going to be) a math major or minor. Math 331 is one of the core courses for the mathematics major. Consequently, I expect that this course will be the one that you devote most of your attention to this term. Students usually need to spend 3 to 5 hours reading and working on problems for each lecture. This should include reviewing class notes, trying to recreate proofs discussed in class, and memorizing key definitions and theorems.

I assume that you have been very successful in Calculus I and II (and that you still remember this material) and that you have had a good exposure to careful proofs through your work in Math 135 and Math 204. The skills that you learned in those two courses will be invaluable here. Also, when we construct the real numbers (yes, construct!), you will encounter some of the structures you worked with in linear algebra and Math 135.

Since there are no lab periods for this course, you must be responsible for your own review and practice. I will suggest lots of extra problems, but it will be up to you to try these and come see me when you are having difficulty. This is why you should be spending 3 to 5 hours of work per lecture on this course.

Definitions and theorems take on new importance in this course. Your familiarity with concepts and previous results in the course will make the new topics in class meetings much more understandable. Spend time learning definitions and the statements of theorems. Try to create your own examples that illustrate them.

Assessment

Homework, reading, and practice exercises will be assigned at the beginning of most classes. I encourage working in small groups on practice problems. This can be very helpful in understanding the material.

About once a week, there will be a graded assignment consisting of selected problems to hand in. Unless otherwise stated, graded assignments are to be your own work without collaboration. These homework assignments should be done neatly. Work out a draft of your solutions on scrap paper first and then make a final copy to hand in. Use pencil rather than pen to avoid having to cross out work. These problem sets when neatly and carefully done are an investment.... you can use them to study for exams. No late assignments, please.

Additionally I expect to ask each of you to do one in-class presentation of a theorem or homework problem. I may also to give some announced, in-class quizzes on definitions or statements of theorems.

In addition to the aforementioned homework and quizzes, there will be three hour tests and a final exam. The dates are listed in the outline below. Tests will be cumulative but will concentrate on more recent material. It is impossible to construct fair makeup exams in mathematics. For your own protection, my policy is that there are no makeup examinations. If for some extraordinary reason you find you are unable to take an exam, let me know as soon as possible, certainly well before the exam is administered.

Your course grade will be calculated as follows.

Homework, Presentation, and any Quiz Grades: 25%
Attendance at a Departmental Seminar: 4%
Three 1-hour exams: 17% each
Final Exam: 20%

I also reserve the right to take class participation and attendance into account in determining final grades.

Attendance and Courtesy

Because of the nature of this course, its assignments, and its assessment, your attendance and participation is crucial. Mathematics is learned by regular, sustained, attentive effort over an extended period. Only when such effort has been invested will the concentrated study for an exam have any benefit. Therefore, attendance at class is required. Unexcused absences may adversely affect your grade. If you must miss a class for some reason beyond your control, talk to me about it in advance.

Finally, common courtesy demands that you be on time for class and that you do not leave the room during class (unless you are ill). This will help you, your classmates, and me to give our full attention to the course.

Tips for Success

My best advice is to take good, complete notes during class. Even if you don't understand every detail during a lecture, with some patience you should be able to review each day's lecture and understand everything we did. If you don't, then you should come to see me as soon as possible. Here are a few simple things that you can do to be more successful in the course

Come to all classes.
Did I mention that you should take good, complete notes? If I think it is important enough to write down, so should you!
Do the readings with pencil in hand. Write up the answers to practice problems carefully. These will be helpful when studying for an exam.
Keep the answers to problems and notes on the readings in a journal.
Review the notes from the last class before coming to class again. In fact, I suggest recopying the notes! This makes studying for a test easy.
Come to see me for help whenever you need it, preferably before you get too far behind.
Start assignments early. Do not wait until just before an assignment is due to get help.
Ask questions in class. Ask questions about the homework and readings.

Outline of Weekly Readings

This assumes a fairly rapid pace through the listed materials. We may need to adjust this schedule based on our actual work in class.

Week	Topics and Readings
Weeks 1 and 2	Construction of the Real Numbers & their Properties. Chapter 1
Weeks 3 and 4	Limits: Chapter 2, Sections 1-3.
Monday, February 16	Exam 1
Weeks 5 and 6	Continuity. Chapter 2, Sections 4 and 5.
Weeks 7 and 8	Differentiation. Chapter 3 Sections 1-3.
Wednesday, March 11	Exam 2
Weeks 9, 10 and 11	Integration. Chapter 3, Sections 4-7
Week 12	Sequences. Chapter 4, Sections 1-2.
Monday, April 20	Exam 3
Week 13	Series. Chapter 4, Sections 3-4.
Week 14 and 15	Sequences and Series of Functions: Chapter 4, Sections 5 and 6.
Saturday, May 9, 2009	Final Exam at 1:30 PM

A Note about the Center for Teaching and Learning (CTL)

Hobart and William Smith Colleges encourages students to seek the academic collaboration and resources that will enable them to demonstrate their best work. Students who would like to enhance their study skills, writing skills, or have other academic inquiries should contact the CTL. You may visit the CTL web site to learn more about the services and programs that are available: http://www.hws.edu/academics/ctl/

If you are a student with a disability for which you need or may need accommodations, you should self-identify and register with the Coordinator of Disability Services at the Center of Teaching and Learning. You will then be required to provide for review documentation of your disability to that office. Disability related accommodations and services generally will not be provided until the registration and documentation process is complete. Guidelines for documenting disabilities and information pertaining to registration with the CTL can be found at our website: http://www.hws.edu/disabilities

If you have a question about this process or Disability services at HWS, please contact David Silver (Coordinator of Disability Services - CTL) at silver@hws.edu or x3351.