Math 4310. Introduction to Real Analysis.

Syllabus for Math 4310 (Introduction to Real Analysis), Fall 2018

TuTh 2-3:15 pm, New Cabell 032.

Instructor: Mikhail Ershov
Office: Kerchof 302
e-mail: ershov at virginia dot edu
Office hours: 3 hrs TBA and by APPOINTMENT
Course webpage: http://people.virginia.edu/~mve2x/4310_Fall2018

Main text (required): Real Mathematical Analysis, Charles Pugh, second edition, corrected printing.

Additional text (recommended): Principles of Mathematical Analysis, Walter Rudin, third edition.
Additional text (strongly recommended to buy): Introductory Real Analysis, A. N. Kolmogorov and S. V. Fomin, revised English edition.

Prerequisites

1. You should have taken MATH 3310 (Basic Real Analysis) or equivalent. The most essential skill from 3310 that is crucial in 4310 is being able to work with the epsilon-delta definition of the limit and related notions like convergence and continuity (just formally understanding the definition of the limit is not enough).
2. As with most 4000 level math classes, you are expected to have substantial prior exposure to proofs. You should be completely comfortable with basic proof techniques and have experience in both reading (and understanding) proofs and writing your own proofs.

Course outline

The main goal of the course is to develop basic notions of mathematical analysis (convergence, continuity, compactness etc.) in the setting of metric spaces. The material in the first half of the course will have some overlap with that of 3310; however, almost everything will be done in greater depth and generality. The second half will cover topics that you most likely have not studied before, including convergence in function spaces, fixed point theorems and Lebesgue integration. Our main textbook will be `Real Mathematical Analysis' by Pugh; however, we will not be following it too closely. I hope to cover the majority of topics from chapters 1,2 and 4 from Pugh's book and some parts of chapters 3 and 6. The book `Introductory Real Analysis' by Kolmogorov and Fomin will be the main reference for Lebesque integration (roughly the last three weeks of the semester).

Evaluation

The course grade will be based on homework, two midterms and the final, with weights distributed as follows:

homework: 30%
midterms: 20% each
final: 30%

Exams

The format of the exams has not been finalized at this point, but most likely the final and the first midterm will be in-class while the second midterm will be take-home. The midterm dates given below are tentative and may be changed later. The date and time of the final exam is determined by the registrar and cannot be changed (in case the exam is held in class).

First midterm exam: Thu, Oct 4th
Second midterm exam: Thu, Nov 8th
Final exam: Mon, Dec 17th, 9am-12pm (the date could be changed in the case of a take-home final)

Make-up policy

Make-ups or extensions (for take-home exams) will be given only under extreme circumstances (such as serious illness). Except for emergencies, you must obtain my permission for a make-up (resp. extension) before the exam (resp. due date).
If you miss an in-class exam or fail to submit a take-home exam by the due date without a compelling reason, you will be assigned the score of 0 on that exam.
University regulations specifically prohibit early make-ups.

Homework

Homework will be assigned weekly and will usually be due on Thursday.
Please STAPLE your homework.
No late homework will be accepted. However, two lowest homework scores will be dropped.
GRADING: it will not be possible to grade all homework problems. In each assignment 3-4 selected (but not announced in advance) problems will be graded for credit.

Collaboration policy.

On homework: you are welcome (and even encouraged) to work on homework together, but you must write up solutions independently, in your own words. In particular, you should not be consulting others during the process of writing down your solution.
On take-home exams: you MAY NOT discuss problems with other people or use any resources (including web) except for the class textbook and your class notes. You may ask me questions about the exam problems, but normally I will only provide very brief hints.

Announcements

Major announcements will be made in class and also posted on the course webpage. Some other announcements may only be made by e-mail, so check your e-mail account regularly.

Add/drop/withdrawal dates:

Tuesday, September 11 -- Last day to ADD a course, elect the audit option, change the grading option (grade or CR/NC), or establish an independent study
Wednesday, September 12 -- Last day to DROP a course (deletion from the transcript)
Tuesday, October 23 -- Last day to withdraw from a course

SDAC

All students with special needs requiring accommodations should present the appropriate paperwork from the Student Disability Access Center (SDAC). It is the student's responsibility to present this paperwork in a timely fashion and follow up with the instructor about the accommodations being offered. Accommodations for test-taking (e.g., extended time) should be arranged at least 5 business days before an exam.