MATH 471 "Analysis I"

Syllabus: short review of Riemann integration, measures (outer measure on R, measurable spaces and functions, convergence of measurable functions, etc.), integration (integration with respect to a general measure, limits of integrals, integrals of limits), differentiation (Hardy-Littlewood maximal function, derivatives of integrals), product measures (iterated integrals, Lebesgue integration on R^n), Banach spaces (metric spaces, vector spaces, normed vector spaces, linear functionals), L^p spaces.

Prerequisites: MATH 265H and, preferably, MATH 240H.

Textbook: Sheldon Axler, Measure, Integration & Real Analysis, Springer Open, Graduate Texts in Mathematics, 2020. A free electronic version, together with an errata and supplement, can be found here.

In addition to the textbook, one should work problems from "Problems in mathematical analysis III - Integration" by W. J. Kaczor and M. T. Nowak, which can be found as an e-book on the website of the University of Rochester libraries.

Course philosophy

This is a rigorous graduate-level course that requires a significant time commitment. Proficiency will be achieved only through diligent work, consistent problem solving, and active participation in class discussions. Please take full advantage of my office hours.

Grading: quizzes (15%), homework (15%), midterm exam (30%), final exam (40%).

Quizzes

Quizzes (10-15 minutes long) will assess understanding of the theoretical concepts covered in class and/or their direct application to an exercise or problem. There will be 10 quizzes from which the best 8 will count toward your grade. There will be no make-up quizzes.

Homework

Homework is usually assigned weekly on Wednesday, starting 9/9, and it is due back the following Thursday by 23:59. There will be 10 assignments from which the best 8 will count toward your grade. Late homework is not accepted.

The homework should be uploaded to Gradescope as a single PDF file.

Exams

The midterm exam will be an in-class exam on Monday, 10/19, and will be based on material covered from the start of the semester all the way to and including the lecture on 10/7. A review session will be held on 10/14.

The final exam, which is a comprehensive exam, will take place on Sunday, 12/20, 12:30-15:30. Review sessions will be held on 12/9 and 12/14.

Course policies

1. The course average is not based on a curve, nor on previously fixed scales. It will reflect how well the class is doing, and it will be high if everyone is working hard for the homework and is performing well on the exams.

2. Incomplete "I" grades are almost never given. The only justification is a documented serious medical problem or a genuine personal/family emergency. Falling behind in this course or problems with workload on other courses are not acceptable reasons.

3. If you miss one of the exams with a valid excuse (e.g., illness or emergency), you must notify the instructor and provide supporting documentation verifying your excuse as soon as possible. For a valid excuse with supporting documentation, the other exam will count as your make-up test. If you miss both exams, you are in danger of failing the class. In principle, no make-up exams will be offered. If you miss an exam without a valid excuse (and supporting documentation), you will receive a score of 0 on that test.

4. You are responsible for knowing and abiding by the University of Rochester's academic honesty policy. Any violation of academic honesty will be pursued according to the specified procedures. Direct copying of answers from A.I. tools is not allowed and considered academic dishonesty. Furthermore, the following Mathematics Department policy also applies to this class:

       Any usage whatsoever of online solution sets or paid online resources 
       (chegg.com or similar) is considered an academic honesty violation and 
       will be reported to the Board on Academic Honesty. In particular, any 
       assignment found to contain content which originated from such sources 
       is subject to a minimum penalty of zero on the assignment and a full letter 
       grade reduction at the end of the semester (e.g. a B would be reduced to a C). 
       This applies even if the unauthorized content was obtained through indirect 
       means (through a friend for instance) and/or the student is seemingly unaware 
       that the content originated from such sources. If you have any questions about 
       whether resources are acceptable, please check with your instructor.

5. This course follows the College credit hour policy for four-credit courses. This course meets 3 academic hours per week. Students may also be expected to deepen their understanding of the course material through close examination/evaluation of the readings assigned in the course.

Tentative weekly schedule