21-455: Intermediate Real Analysis

Fall 2024

Course Information

Homework

Homework is due every Wednesday at 12:00 (noon) on Gradescope. (Use this invite code if you’re not signed up). Please read the late homework policy

Handouts

Office Hours and Discussion Board

  • Use the discussion board for all questions.
  • Join to the mailing list to receive announcements.
  • Gautam Iyer’s office hours: TBA (in WEH 8115).
  • Eric Wang’s office hours: TBA (in WEH 7106)

Exam Schedule

  • Midterm 1: Wed, Sep 25 (in class)
  • Midterm 2: Wed, Nov 13 (in class)
  • Final: TBA

Syllabus

Course Description

This is a second course on analysis, and will cover various topics in Metric spaces:

  • Metric spaces: Completeness, density, separability, compactness, connectedness.
  • Baire theorem and applications.
  • Contraction maps: Fixed points, inverse, and implicit function theorems.
  • Spaces of functions: Uniform and pointwise convergence, Stone-Weierstrass theorem, Arzela-Ascoli theorem.

Learning Objectives

  • Understand the basics of metric spaces, which are a fundamental tool in modern analysis.
  • Develop essential mathematical skills such as reading / writing proofs, constructing counter examples, and problem solving.

Pre-requisites

  • A rigorous and through first course on Calculus / analysis (e.g. 21-269)
  • Familiarity with reading / writing proofs in an abstract setting.

References

There is no textbook for this class. Here are a few references; you’re free to follow any one that resonates with your style.

  • Brief Notes containing only definitions, and statements of results we will cover in class. These notes do not have any proofs; we will do the proofs in class, and on your homework.
  • Principles of Mathematical Analysis by W. Rudin
  • Functions of Several Variables by Fleming

Grading

Your scores on exams and homework will each be converted to a scaled score with A=90, B=80 etc. using a rubric that will be announced after each exam. The scaled scores will be averaged with the following weights:

  • Each midterm will count as 20%.
  • The final will count as 30%.
  • Homework will count as 30% of your grade.

In order to pass the class you must the exams. Precisely, your average scaled score on exams, without counting your homework, and with weights proportional to those listed above, must be at least 60, corresponding to a D.

Class Policies

Lectures

  • If you must sleep, don’t snore!
  • Be courteous when you use mobile devices

Homework

  • All homework must be scanned and turned in via Gradescope. (Everyone who was registered for this class on day 1 was automatically added to Gradescope. If you joined later, use this invite code to add yourself.)
  • Please take good quality scans; homework that’s too hard to read won’t be graded. I recommend using a good scanning app that adjusts the contrast of your images for readability. (I’ve had good luck with Adobe Scan, and Google Drive.)
  • I recommend starting the homework early. Most students will not be able to do the homework in one evening.
  • You may collaborate, use books and whatever resources you can find online in order to do the homework. However, you must write your solution up independently, and you must fully understand any solution you turn in. Turning in solutions you don’t understand will be treated as a violation of academic integrity.
  • In order to ensure academic integrity is maintained, I will call on some subset of students to explain their solutions to me outside class.
  • New material will be developed through homework problems. If you are unable to solve a particular problem, be sure to ask me or your TAs about it, or look up the solutions after they are posted.
  • Some homework problems will also appear on your exams with a devious twist. A through understanding of the solutions (even if you didn’t come up with it yourself) will invariably help you. But knowledge of the solution without understanding will almost never help you.
  • Nearly perfect student solutions may be scanned and hosted here, with your identifying information removed. If you don’t want any part of your solutions used, please make a note of it in the margin of your assignment.

Exams

  • All exams are closed book, in class.
  • No calculators, computational aids, or internet enabled devices are allowed.
  • The final time will be announced by the registrar here. Be aware of their schedule before making your travel plans.

Academic Integrity

  • All students are expected to follow the academic integrity standards outlined here.
  • There will be zero tolerance for academic integrity violations, and any violation will result in an automatic R. Examples of academic integrity violations include (but are not limited to):
    • Not writing up solutions independently and/or plagiarizing solutions.
    • Turning in solutions you do not understand.
    • Receiving assistance from another person during an exam.
    • Providing assistance to another person taking an exam.
  • All academic integrity violations will further be reported to the university, and the university may chose to impose an additional penalty.

Accommodations for Students with Disabilities

If you have a disability and have an accommodations letter from the Disability Resources office, I encourage you to discuss your accommodations and needs with me as early in the semester as possible. I will work with you to ensure that accommodations are provided as appropriate. If you suspect that you may have a disability and would benefit from accommodations but are not yet registered with the Office of Disability Resources, I encourage you to contact them at access@andrew.cmu.edu.

Student Wellness

As a student, you may experience a range of challenges that can interfere with learning, such as strained relationships, increased anxiety, substance use, feeling down, difficulty concentrating and/or lack of motivation. These mental health concerns or stressful events may diminish your academic performance and/or reduce your ability to participate in daily activities. CMU services are available, and treatment does work. You can learn more about confidential mental health services available on campus here. Support is always available (24/7) from Counseling and Psychological Services: 412-268-2922.

Faculty Course Evaluations

At the end of the semester, you will be asked to fill out faculty course evaluations. Please fill these in promptly, I value your feedback. As incentive, if over 75% of you have filled out evaluations on the last day of class, then I will release your grades as soon as they are available. If not, I will release your grades at the very end of the grading period.