Math 521 Analysis I, Spring 2016

Our subject this semester

Real Analysis deals with the notion of limits and convergence, either of numbers, or vectors, or sequences of functions. To deal with “convergence” in these different contexts we will introduce the theory of metric spaces early on in the semester. Following that we study convergence of sequences and series of real numbers and vectors. We then revisit the theory of differentiation and integration of functions of one variable. Our last topic will be convergence of sequences of functions, where one can distinguish between pointwise convergence, uniform convergence, convergence in the mean. See also Math Department page for Math 521. In Rudin's book this material is in Chapters 1-7.

Instructor.

Sigurd Angenent, 609 Van Vleck Hall.

Textbook:

Walter Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill.

Grades and exams

Homework (see below) will be assigned weekly. There will be three in-class midterm exams and one cumulative final exam.

Grading scheme

Homework 10%, midterm exams 20% each, final exam 30%. 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%.

Exam Dates

Office hours, Email, Piazza

Email is not a good medium for mathematical discussion, and as the enrollment in our class has reached 40, office hours are not an effective way for answering mathematical questions about differential equations. Instead, please direct all your questions to the Piazza page for our course. I will check Piazza daily. You can post questions, answer other students’ questions, chime in (“I had the same question”), and browse old questions and answers. You can post questions anonymously to the other students, if you like.

Weekly Lecture Schedule

Chapters and sections refer to Rudin’s textbook.
Week 1: Chapter 1. The real number system, extended reals, Euclidean space.
Week 2: Chapter 1. Schwarz inequality in $\R^n$. Chapter 2. Countable and uncountable sets, metric spaces.
Week 3: Chapter 2. Properties of open and closed sets. Compact sets in a metric space.
Week 4: Chapter 2. Compact sets in a metric space. Connected sets.
Week 5: Chapter 3. Sequences, subsequences, Cauchy sequences.
Week 6: Chapter 3. Monotone sequences, limsup and liminf.
Week 7: Chapter 3. Series.
Week 8: Chapter 4. Limits and continuity for functions.
Spring break March 21-25.
Week 9: Chapter 4. Discontinuities of monotone functions. Chapter 5. Derivative.
Week 10: Chapter 6. Riemann-Stieltjes integral: definition and basic properties.
Week 11: Chapter 6. Riemann-Stieltjes integral: criteria for integrability, step function integrator.
Week 12: Chapter 6. Riemann-Stieltjes integral: differentiable integrator, change of variables. Differentiation and integration.
Week 13: Expectations of random variables as Riemann-Stieltjes integrals. Informal discussion of Riemann versus Lebesgue integration. Chapter 7. Pointwise and uniform convergence of sequences of functions.
Week 14: Chapter 7. Compactness in $C(X)$. Weierstrass approximation theorem.
Week 15: Chapter 7. Finish Weierstrass approximation theorem. Separability of $C([a,b])$. Chapter 4. Intermediate Value Theorem.

Homework

Weekly homework assignments will be posted here. 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:

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.