Honors: Introduction to Analysis I & II

**MATH 5615H Section 001, Fall 2017, MWF 12:20-1:10 in Vincent Hall 2**

**Instructor**: Douglas N. Arnold, arnold@umn.edu

**Office hours**: Monday 11:30-12:15, Wednesday and Friday 1:15-2:00 in Vincent Hall 512, or by appointment

**Lecture schedule**
• **News and announcements** (see here for homework assignments)

**Course description**: MATH 5615H-5616H is a two-semester introduction to real analysis at the honors undergraduate level. The first semester will revisit basic properties of the real numbers, sequences and series of real numbers, and functions of a real variable, culminating with the derivative and the Riemann integral. Although much of this material will be familiar to students from their calculus classes, in MATH 5615H we will study it at a much deeper level, with an emphasis on mathematical rigor and foundations. In fact we will start by assuming only the most basic properties of the natural numbers and build up to the real numbers from there. In the second semester course, MATH 5616H, we shall treat more advanced topics such as metric spaces, power series, Fourier series, multivariable calculus, and, finally, the Lesbesgue integral. An important aspect of the course will be learning to read, understand, appreciate, and write rigorous mathematical theorems and proofs.

**Textbook**: The course will closely follow the textbook *Analysis I*, Third Edition for 5615H, and *Analysis II*, Third Edition for 5616H. Both books are authored by Terence Tao. When accessed from UMN IP addresses, PDFs of both books may be freely downloaded at the links just given. When coming from other IP addresses, use these links (which require UMN authentication):
*Analysis I* and
*Analysis II*. It is highly recommended that students purchase printed copies of the text. Hardcover editions may be purchased for under $20 for each book from Amazon. (Pay attention to the shipping time, when ordering.)

**Homework**: The exercises in the text are an integral part of the material, and much of the learning will come by working through them consistently and studiously. Many of these will be assigned and a significant portion graded. Most consist of filling in details of proofs from the text, or writing complete proofs. They will be graded for validity, clarity, and style. Homework is due at the start of class on the day it is due. Late homework will normally not be accepted.

**Exams**: There will be two in-class midterm exams, to be held on Wednesday 10 October and Wednesday 14 November. The final exam will be on Saturday 15 December from 10:30-12:30. Make-up exams are only possible for verified legitimate reasons as defined by the University policy on makeup work, and then only *before* the actual exam, except when unavoidable.

**Grading**: Grades will be based on the homework (25%), the midterm exams (20% each), and the
final exam (35%), and assigned in compliance with the University policy on grading.

**Policies**: All UMN students are expected to abide by the Student Conduct Code and the policy on Student Responsibilities. Students are welcome to discuss the course content with each other and with me, but are expected to do their own homework. This can be a fine line to parse, but at a minimum you should try to solve a problem on your own before consulting with anyone else, and you must write out your solutions yourself in your own words. You should be able to demonstrate your understanding by reproducing the solution later on your own (which you will be tested on).

Other university policies relevant to all classes:

- Sexual harrassment, sexual assault, stalking and relationship violence
- Equity, Diversity, Equal Employment Opportunity, and Affirmative Action
- Disability Services
- Academic Freedom and Responsibility
- Resources related to mental health, stress management, and counseling

Updated September 17, 2018