Math 73/103
Metric Spaces and Measure Theory
General Information
Instructor: Danie van Wyk
E-Mail: danie.w.van.wyk-at-dartmouth.edu
Office: 314 Kemeny Hall
Office Hours: Tu 2-3:30 p.m., W 10-11 a.m.
Lectures: MW 15:30-17:20 p.m. in 006 Kemeny Hall
x-hour: 5:30-6:20 p.m.
During the x-hour I will be usually available in my office, but x-hours will be occasionally used for class work.
Content
Note: Math 73/103 no longer covers complex analysis, which is replaced with metric spaces.
This course is an introduction to graduate level analysis. The first part of the course covers metric spaces. In this part we will introduce metric spaces and the metric topology. We'll discuss completeness and show how to complete a metric space. Then we will study compact metric spaces and work toward proving two important results, namely the Arzela-Ascoli Theorem and the Baire Category Theorem.
The second part of this course covers abstract measure theory. This part starts with the introduction of measurable sets and measures on spaces. This will allow us to define the Lebesque integral which has a number of advantages over the Riemann integral. We will then take a look at \(L^1 \) spaces which are very useful in functional analysis and then end this part with the proof of the Radon-Nikodym theorem.
A prior knowledge of basic set theory, topology and real analysis is required for this course. Additionally you should also be familiar with complex numbers.
Textbooks
We do not have a prescribed textbook. The following books are recommendations, but not required:
Real and Complex Analysis (Third edition) by W. Rudin
Real Analysis (Fourth edition) by H.L. Royden and P.M. Fitzpatrick
Real Analysis, Modern techniques and their applications (Second edition) by G.B. Folland
Exams
The exams are scheduled as follows:| Midterm Exam | October 20 | Take home exam |
| Final Exam | TBD |
If you have a conflict with the midterm exam because of a religious observance, scheduled extracurricular activity such as a game or performance [not practice], scheduled laboratory for another course, or similar commitment, please see your instructor as soon as possible.
Grades
| Written homework | 30% |
| Oral presentation | 10% |
| Midterm exam | 30% |
| Final exam | 30% |
COVID-19 Information
Attendance:You are expected to attend class in person unless you have made alternative arrangements due to illness, medical reasons, or the need to isolate due to COVID-19. For the health and safety of our class community, please: do not attend class when you are sick, nor when you have been instructed by Student Health Services to stay home.
Safety:In accordance with current College policy, all members of the Dartmouth community are required to wear a suitable face covering when indoors, regardless of vaccination status. This includes our classroom and other course-related locations, such as labs, studios, and office hours. If you need to take a quick drink during class, please dip your mask briefly for each sip. Eating is never permitted in the classroom. (The only exception to the mask requirement is for students with an approved disability-related accommodation; see below.) If you do not have an accommodation and refuse to comply with masking or other safety protocols, I am obligated to assure that the Covid health and safety standards are followed, and you will be asked to leave the classroom. You remain subject to course attendance policies, and dismissal from class will result in an unexcused absence. If you refuse to comply with masking or other safety protocols, and to ensure the health and safety of our community, I am obligated to report you to the Dean’s office for disciplinary action under Dartmouth’s Standards of Conduct. Additional COVID-19 protocols may emerge. Pay attention to emails from the senior administrators at the College.
Student Accessibility Services
Students requesting disability-related accommodations and services for this course are required to register with Student Accessibility Services (SAS; Getting Started with SAS webpage; student.accessibility.services@dartmouth.edu; 1-603-646-9900) and to request that an accommodation email be sent to me in advance of the need for an accommodation. Then, students should schedule a follow-up meeting with me to determine relevant details such as what role SAS or its Testing Center may play in accommodation implementation. This process works best for everyone when completed as early in the quarter as possible. If students have questions about whether they are eligible for accommodations or have concerns about the implementation of their accommodations, they should contact the SAS office. All inquiries and discussions will remain confidential.
Recordings
(1) Consent to recording of course and group office hours: By enrolling in this course, a) I affirm my understanding that the instructor may record this course and any associated group meetings involving students and the instructor, including but not limited to scheduled and ad hoc office hours and other consultations, within any digital platform used to offer remote instruction for this course; b) I further affirm that the instructor owns the copyright to their instructional materials, of which these recordings constitute a part, and my distribution of any of these recordings in whole or in part without prior written consent of the instructor may be subject to discipline by Dartmouth up to and including expulsion;
(2) Requirement of consent to one-on-one recordings: By enrolling in this course, I hereby affirm that I will not under any circumstance make a recording in any medium of any one-on-one meeting with the instructor without obtaining the prior written consent of all those participating, and I understand that if I violate this prohibition, I will be subject to discipline by Dartmouth up to and including expulsion, as well as any other civil or criminal penalties under applicable law.
The Honor Principle
Academic integrity is at the core of our mission as mathematicians and educators, and we take it very seriously. We also believe in working and learning together.
Collaboration on homework is permitted and encouraged, but obviously it is a violation of the honor code for someone to provide the answers for you.
On written homework, you are encouraged to work together, and you may get help from others, but you must write up the answers yourself. I want to hear your interpretation of the material.
On exams, you may not give or receive help from anyone.
Exams in this course are closed book, and no notes, calculators or other electronic devices are permitted.Religious Observances
Some students may wish to take part in religious observances that occur during this academic term. If you have a religious observance that conflicts with your participation in the course, please meet with me as soon as possible, or before the end of the second week of the term—at the latest, to discuss appropriate adjustments.
Mental Health
Even without the global pandemic, the academic environment at Dartmouth is challenging, our terms are intensive, and classes are not the only demanding part of your life. In the midst of a global pandemic, with all the uncertainty surrounding every aspect of our lives, these challenges take on an extra toll. There are a number of resources available to you on campus to support your wellness, including your undergraduate dean. Counseling and Human Development and the Student Wellness Center.