## Syllabus for Math 4310 (Introduction to Real Analysis), Fall 2015

### TuTh 2-3:15 pm, Dell 2 103.

• Instructor: Mikhail Ershov
• Office: Kerchof 302
• e-mail: ershov at virginia dot edu
• Office hours: 2 hrs 30 min TBA and by APPOINTMENT
• Course webpage:   http://people.virginia.edu/~mve2x/4310_Fall2015
• Main text (required): 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 `Principles of Mathematical Analysis' by Rudin; however, we will not be following it too closely. The tentative plan is to cover chapters 1,2,4,7,11 from Rudin's book and some parts of other chapters. 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 the case of a take-home final it will be due on that date).
• First midterm exam: Thu, Oct 1st
• Second midterm exam: Thu, Nov 5th
• Final exam: Tue, Dec 15th, 9am-12pm

### 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.
• 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.