Math 521 Analysis I, Spring 2017

Lectures

This page is for lectures 001 and 002 of math 521 in spring 2017. Sigurd Angenent is the instructor.

The textbook

We will use Rudin’s “Principles of Mathematical Analysis,” third edition. I will assign some homework from this book, and also post other homework problems on this website. The UW has a page with Roger Cooke's solutions to the problems in Rudin’s book

At times I will deviate from the textbook. The following pages provide notes on the topics presented in lecture.

	Su	Mo	Tu	We	Th	Fr	Sa
Jan				18	19	20	21
	22	23	24	25	26	27	28
Feb	29	30	31	1	2	3	4
	5	6	7	8	9	10	11
	12	13	14	15	16	17	18
	19	20	21	22	23	24	25
March	26	27	28	1	2	3	4
	5	6	7	8	9	10	11
	12	13	14	15	16	17
	SPRING BREAK
		27	28	29	30	31	1
April	2	3	4	5	6	7	8
	9	10	11	12	13	14	15
	16	17	18	19	20	21	22
	23	24	25	26	27	28	29
May	30	1	2	3

Exams and grades

There will be two midterm exams:

midterm 1: Wednesday, February 15
midterm 2: Wednesday, March 29

These will be evening exams. The final exam will be held as scheduled in the time table.

Grades will be based on homework, midterm, and final exam scores. The midterms count for 25% each, the final will count for 30%, and the homework will count for the remaining 20%. A letter grade will be determined according to this tentative set of cut offs:

A≥92%, AB≥86%, B≥78%, BC≥70%, C≥60%, D≥50%, F≥0%.

Email, online office hours, and Piazza

Our Piazza page:

    https://piazza.com/wisc/spring2017/math521/home

Instead of in–person office hours we will use Piazza to communicate questions outside of class. Before the class starts students will receive an email inviting them to sign up for piazza. Students can post questions, anonymously if they prefer, discuss and answer other students’ questions. I will moderate the online discussions, and provide hints towards solutions, if necessary.

Do not send me email: it will get lost in the other mail I get. Instead post questions on Piazza.

Course Outline

Real numbers

Fields; ordered fields; inequalities.
The Least Upper Bound Axiom and the real number field $\R$.
Complex numbers.
Euclidean spaces and Cauchy’s inequality.

Metric spaces

Set theory. Finite, and infinite sets. Countable and uncountable sets.
Metric spaces. Open and closed subsets. Neighborhoods and limit points.

Sequences and series of real numbers

In class we used the definition of compactness involving convergent subsequences. This deviates from Rudin’s approach. A short write-up of the definitions and theorems can be found here.

Continuity

Differentiable Functions

The Riemann integral

Sequences and series of functions

Homework

Weekly homework assignments will be posted on the homework page. They are due in class on Monday.

Academic Integrity. You are encouraged to exchange thoughts with other students, but in the end you must write and hand in your own personal solutions. Solutions (some correct, some not) to many problems in Rudin’s book are easily Googled. Handing in plagiarized work, whether copied from a fellow student or off the web, is not acceptable. See the UW website on Academic integrity.

Homework is collected in class each Monday. Late papers will not be accepted for any reason.

What should homework look like? Solving math problems is done in two stages. First you figure out the solution. This involves lots of cheap scratch paper, erasers, and a trash bin. Once you have figured out the solution, you write it up. Keep the following points in mind when you write up your solution:

Use proper English: write in complete English sentences. Mathematical formulas should be embedded in complete sentences.
Your solutions should make sense when read from top to bottom. The grader will interpret your solution according to what you wrote, not what you meant.
Explain all notation you use if it is not in the textbook.
Practice being concise: is there a shorter way to write what you just wrote?

Finally, about the physical form of the work you hand in: staple your pages together and put the problems in correct order. Do not use paper torn out of a binder. Be neat: there should not be things crossed out or ugly eraser marks. Messy or unreadable papers cannot be graded.

“How much detail should I include?” You will invariably run into a situation where you are not sure about whether some fact can be taken for granted or whether it needs to be proved. In that case the safe alternative is to give a justification. You can put such technical lemmas in an appendix at the end of your solution to avoid interrupting the main flow of your argument. Rule of thumb: if the grader needs pencil&paper to check something you claim, you should have proved it.