Math 521 Analysis I, Lecture 2, Spring 2011

Final Exam on Saturday May 14, 7:45-9:45 AM, in Social Science 6102. The exam covers the entire course. Please bring pencils and scrap paper, no calculators or other electronics.

Meetings: Van Vleck B235, 12:05-12:55 MWF

Instructor: Timo Seppäläinen

Office: 419 Van Vleck, Office Hours: MW 10-11 or by appointment

Phone: 263-2812

E-mail: seppalai@math.wisc.edu

Material

In Math 521 we learn to handle rigorously familiar ideas from calculus such as limits, continuity, differentiation and integration. Reading and writing proofs and producing counterexamples to false claims are central to the course. There is also an emphasis on precise use of terminology and problem solving without relying on the textbook. That is why books, notes or calculators will not be allowed in exams or quizzes.

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

You can see the list of topics to be covered in the Mathematics Department's information page for Math 521. In Rudin's book this material is in Chapters 1-8.

Grades and exams

The table below gives tentative grade lines. In the end these grade lines may need to be adjusted to reflect the actual performance of the class in relation to historical grades.

Grade	A	AB	B	BC	C	D
Percentage	90	84	78	72	65	55

Tentative Exam Dates:	Exam 1	Friday, February 11 (week 4)
	Exam 2	Wednesday, March 9 (week 8)
	Exam 3	Wednesday, April 13 (week 12)
	Final Exam	Saturday, May 14, 7:45-9:45 AM

Weekly schedule

This schedule tracks our progress in Rudin's text.

• Week 1. Chapter 1. The real number system, extended reals, Euclidean space.
• Week 2. Chapter 1. Schwarz inequality in Rⁿ. Chapter 2. Countable and uncountable sets, metric spaces.
• Week 3. Chapter 2. Properties of open and closed sets. Compact sets in a metric space. Quiz 1 on Friday. Solution.
• Week 4. Chapter 2. Compact sets in a metric space. Exam 1 on Friday. Solution.
• Week 5. Chapter 3. Sequences, subsequences, Cauchy sequences.
• Week 6. Chapter 3. Monotone sequences, limsup and liminf.
• Week 7. Chapter 3. Series. Quiz 2 on sequences on Wednesday. Solution.
• Week 8. Chapter 4. Limits and continuity for functions. Exam 2 on Wednesday on sequences and series. Solution.
• Spring break week March 14-18.
• 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. Review for Exam 3.
• Week 12. Chapter 6. Riemann-Stieltjes integral: differentiable integrator, change of variables. Differentiation and integration. Random variables. Exam 3 on Wednesday on continuity and differentiation. Solution.
• 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.
• Solution for the final exam.

Homework

Homework assignments will be posted here. Please check this list before you start working to see the latest updates. Future assignments are tentative and will be adjusted to match progress in class.

• Homework 1 due Monday, January 24. Ch. 1 Problems 2, 3, 5, 6. For practice in mathematical induction: prove that 1³+ 2³+… + n³ = (1+2+…+n)² for all positive integers n. Solution.
• Homework 2 due Monday, January 31. Ch. 1 Problems 15 (in the context of Euclidean space, not complex plane), 17. Ch. 2 Problems 4, 5, 11. Solutions and more solutions.
• Homework 3 due Monday, February 7. Ch. 2 Problems 7, 9, 22, 29. The outcome of problem 29 is sometimes quite useful.
  Solution: page 1 page 2 page 3
  Suggested but not to be handed in at this time: Ch 2: 6, 10, 12, 14, 15, 16.
• Homework 4 due Wednesday, February 23. Ch. 3 Problems 14(a)(b), 20, and this problem from p. 51: use the definition of the limit to show that, in a metric space, a sequence {p_n} converges to p if and only if every subsequence of {p_n} converges to p. In problem 14 assume the sequence is real. Solutions.
• Homework 5 due Monday, February 28. Click here. Solutions.
• Homework 6 due Monday, March 7. Ch. 3 Problems 5, 6(a)(b), 7, 14(c)(d), 16(a). This seems like a lot but the problems are quick if you appeal to theorems, except 16(a) where you need to work a little more. Hints: For 5 it's best to use some theorems instead of the definition of limsup. In 6 note that the trick for finding the asymptotics of √(n+1) - √n is to multiply numerator and denominator by √(n+1) + √n. In 7 use Schwarz inequality. In 14 you can of course appeal to the parts you already proved in the previous homework. 16 gives an approximation to the square root. Start by proving inductively that x_n>√α. To identify the limit you can take n to infinity in the inductive equation between x_n+1 and x_n but this step should be justified (use some theorem(s)). Solutions.
• Homework 7 due Monday, March 28. Ch. 4 Problems 4, 11. The second part of 11 is a pretty important result. Specifically, what you need to show is this: Let E be a dense subset of X and f:E→R uniformly continuous. Then there exists a continuous g:X→R such that g=f on E. Start by showing that if x_n and y_n are sequences in E that converge to a common limit x∈ X, then the sequences f(x_n) and f(y_n) converge to the same limit in R. Then use such sequences to define the values g(x). Argue precisely why your g is well-defined on all of X, and then show g is continuous. Additionally, show that the result is not true without the assumption of uniform continuity. For this it is enough to remove a point from an interval and think of two different constants on the two disjoint pieces. Solutions.
  Suggested but not to be handed in at this time: Ch 4: 1, 2, 3, 8, 10, 12, 20, 21, 23.
• Homework 8 due Monday, April 4. Ch. 5 Problems 8, 11, 26. Solution: page 1 page 2.
  Suggested but not to be handed in at this time: Ch 5: 1, 2, 3, 5, 9, 14.
• Homework 9 due Wednesday, April 20. Ch. 6 Problems 2, 4, 8, and this extra problem. Solution: page 1 page 2.
  Suggested but not to be handed in at this time: Ch 6: 1, 11.
• Homework 10 due Wednesday, May 4. Ch. 7 Problems 9, 18, 20 and this extra problem. Solution: page 1 page 2.
  Suggested but not to be handed in at this time: Ch 7: 1, 2, 3, 6, 7, 16.

Bonus problems

Bonus problems are for students who have signed up for honors credit, and for anyone else with appetite for extra challenge.

• Prove carefully that the closed interval [0,1] is uncountable by using the fact that the set of sequences of zeroes and ones is uncountable. Note that some reals have two representations as series in powers of 1/2.
• Exercises 2.25 and 2.26. The outcomes of these two exercises are very basic for analysis and come in handy a great deal.
• Which sets in Rⁿ are both open and closed?
• Exercises 3.23 and 3.24. These show that every metric space X can be completed in an abstract way: there is a complete metric space Y and a one-to-one map of X into Y such that the image of X is dense in Y. Show also that any two completions are isometric (that is, there is a distance-preserving bijection between them).
• Exercises 4.20-22.
• Exercises 5.15 and 5.22.

Instructions for homework

• Observe rules of academic integrity. Handing in plagiarized work, whether copied from a fellow student or off the web, is not acceptable. Plagiarism cases will lead to sanctions. You are encouraged to exchange thoughts with other students, but in the end you must write and hand in your own personal solutions.
• Homework is collected in class on the due date. Alternately, you can bring it to the instructor's office or mailbox by 3 PM of the due date. No late papers will be accepted for any reason.
• Use proper English. Write in complete English or mathematical sentences. The grader is not in class, hence define all notation you use that is not in the book.
• Staple your pages together. Put problems in correct order. Don't use paper torn out of a binder. Be neat: there shouldn't be things crossed out or ugly eraser marks. Recopy your problems: don't hand in your first attempt that looks like scratch paper. Messy or unreadable papers cannot be graded.
• If you prefer, you are welcome to typeset your homeworks in Latex.
• 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 that sometimes helps: if the grader needs to pick up a pencil to check something you claim, you should have proved it.