Math 331:
Foundations of Analysis

   Department of Mathematics and Computer Science
   Hobart and William Smith Colleges

   Fall 2020.

   Instructor:  David J. Eck  (eck@hws.edu)
   
   Monday, Wednesday, Friday, 1:20–2:20 PM
        Room Eaton 111.

Please note that this is a tentative syllabus, based on the Colleges' plans for the Fall semester as of mid-July. Changes might be necessary as plans become more definite or as the situation changes. In any case, however, the course should cover the material that is listed here.

About The Course and the Textbook

Math 331 is a first course in real analysis. For much of the course, we will be talking about topics that you covered in Calculus I and Calculus II: limits, continuity, differentiation, integration, and infinite series and sequences. It is, however, decidedly not just another calculus course. In your calculus courses, you mostly learned some techniques and used them to solve some problems. For example, you memorized a list of rules for differentiation, and you used them to find the derivatives of a lot of functions (often given by formulas that are unlikely ever to turn up in a real-world application). You might have done some simple proofs, and you probably spent some time developing intuition about why certain things are true, but for the most part, "calculus" meant "calculating" based on some memorized formulas.

In this course, you will do very little calculating. Instead, you will be doing mathematics. That is, we will start from first principles and build up a theory from there, based on rigorous logical reasoning. The first principles are the definition and basic properties of the real numbers. The theory that we will develop is the real Calculus, the theory of the infinite and the infinitesimal that has taken some 2500 years to develop.

The textbook for the course is Foundations of Analysis, Second Edition, by David Belding and Kevin Mitchell. The two authors were professors at Hobart and William Smith Colleges until their retirement in 2017. This book was written specifically as a textbook for a pair of courses that they developed, Foundations of Analysis I and II. We will cover the material of the first of those courses, chapters 1 through 4 of the book.

The textbook was written before Math 135 was invented, and the courses based on it were often the first experience that students had with rigorous mathematics. You should be more prepared for the rigor than those earlier students were. I plan to take advantage of that fact by adding some material on metric spaces, which are often covered in similar analysis courses. We will see how fundamental properties of the real numbers can be generalized to a more abstract setting. The readings for this part of the course can be found online at https://math.hws.edu/eck/metric-spaces/.

About the Pandemic

Dealing with the pandemic will be an issue throughout the semester. While I hope that we will all be able to meet in person and that none of us will get sick or need to be quarantined, only time will tell how realistic that hope may be.

We should expect, according to school policy, to be wearing masks and practicing social distancing in class. No one who is showing symptoms should come to class. It is possible that someone will be asked to self-isolate because of possible exposure to the virus; the policy on that will be set by the Colleges. All of this applies to me as well as to students. In the worst case, we might have to deal with a full shutdown of the Colleges and a transition to remote classes. Detailed policies will be set by the Colleges.

I will try to make the class accessible to students who can't always be there in person. I plan to post reading guides and occasional short videos to supplement the readings from the textbook. I will set up appointment times for individual and group meetings on Zoom. And of course I will always be available on email. If I can't be in class myself, I expect the course to continue either remotely or with a guest lecturer filling in for me.

Assignments

Homework will be assigned and collected weekly. I hope that most people in the course will be willing to write their solutions using LaTeX at overleaf.com. If your work is done at overleaf.com, and if you give me access to your project there, I will be able to view and comment on your work online.For more information about submitting homework, see

http://math.hws.edu/eck/math331/f20/submitting-homework.html

Note that a majority of the homework assignments will involve proving things. In most cases, I will give you feedback on your proofs and a chance to revise them to raise your grade.

You are encouraged to discuss homework assignments with me both before turning them in and after receiving feedback. It is possible that I will ask to meet with you about your revisions.

You are allowed and even encouraged to discuss homework problems with other students in the class. However, you should always write up your own solutions, with your own reasoning and in your own words.

Tests and Final Evaluation

I plan to give two in-class tests that will cover mostly definitions and basic concepts, including some straightforward proofs. The two in-class tests are planned for Monday, October 5 and Friday, November 20. It is not entirely clear what the protocol will be for doing tests on paper, but I'm hoping that it will be possible. There will also be a take-home midterm exam, which will be distributed around the middle of October.

Current planning calls for the end of the course, including the final exam, to be run remotely. I don't yet know how that will work, but I am leaning towards a final summary problem set and an individual Zoom meeting where you can present and discuss your solutions. This is something that we can discuss.

Grading

Here is the approximate weighting for the various components of the course:

             First in-class test:      15%
             Second in-class test:     15%
             Take-home midterm:        15%
             Final evaluation:         15%
             Homework:                 40%

About Office Hours

The Colleges' opening plan advises against having in-person meetings with students in Faculty offices. Since my office is large, it might be possible for me to meet with one person there, but only by appointment, since it will not be possible to hold open office hours in my office. I will schedule some open office hours on Zoom. I will also set up times for individual or group appointments on Zoom. Appointments will be made using the Calendar feature in Canvas. It might be possible to schedule in-person appointments at other venues, such as outdoors. It might even be possible for me to hold open office hours outdoors. Details will be announced.

Of course, email is always a good way to contact me. My email address is eck@hws.edu. I welcome comments and questions by email, and I will usually respond to them fairly quickly.

Statements from the Center for Teaching and Learning

At Hobart and William Smith Colleges, we encourage you to learn collaboratively and to seek the resources that will enable you to succeed. The Center for Teaching and Learning (CTL) is one of those resources: CTL programs and staff help you engage with your learning, accomplish the tasks before you, enhance your thinking and skills, and empower you to do your best. Resources at CTL are many: Teaching Fellows provide content support in 12 departments, Study Mentors help you manage your time and responsibilities, Writing Fellows help you think well on paper, and professional staff help you assess academic needs.

Disability Accommodations: If you are a student with a disability for which you may need accommodations, you should self-identify, provide appropriate documentation of your disability, and register for services with Disability Services at the Center for Teaching and Learning (CTL). Disability related accommodations and services generally will not be provided until the registration and documentation process is complete. The guidelines for documenting disabilities can be found at the following website: www.hws.edu/academics/ctl/disability_services.aspx. Please direct questions about this process or Disability Services at HWS to Christen Davis, Coordinator of Disability Services, at ctl@hws.edu or x 3351

Tentative Schedule

Here is a very tentative schedule for the course. Readings are from the textbook, except for those about metric spaces.

Dates Topics and Readings
Aug. 24, 26, 28 The Real Numbers; Dedekind cuts and the least upper bound axiom.
Sections 1.0–1.2.
Aug. 31; Sept. 2, 4 Axioms for the real numbers; the Heine-Borel Theorem.
Sections 1.3, 1.4.
Sept. 7, 9, 11 Limits, Sections 2.0–2.3.
Sept. 14, 16, 18 Continuity, Sections 2.5, 2.6.
Sept. 21, 23, 25 Metric spaces: Introduction, Open and Closed Sets, and Subspaces.
Sept. 28, 30; Oct. 2 Infinite Sequences and their limits, Sections 4.0, 4.1.
Metric spaces: Limits and Continuity
Oct. 5, 7, 9 The derivative, Sections 3.0, 3.1.
First In-class Test tentatively scheduled for October 5.
Oct. 12, 14, 16 Laws of differentiation; Mean Value Theorem.
Sections 3.2, 3.3.
Take-home Test should be handed out this week.
Oct. 19, 21, 23 The Riemann Integral, Sections 3.4, 3.5.
Oct. 26, 28, 30 Fundamental Theorem of Calculus; Taylor Polynomials.
Sections 3.6, 3.7.
Nov. 2, 4, 6 Monotone and cauchy sequences; Infinite series.
Sections 4.2, 4.3.
Nov. 9, 11, 13 Absolute and Conditional Convergence; Sequences of Functions.
Sections 4.4, 4.5.
Nov. 16, 18, 20 Series of functions, Section 4.6.
Second In-class Test tentatively scheduled for November 20.
Nov. 23 Metric spaces: Compactness.
Nov. 30; Dec. 2 Metric spaces: Completeness.
Metric spaces: Connectedness.
TBA Final Evaluation of some sort