Math 321 - Real Variables II - Spring 2019

Instructor: Malabika Pramanik
Office: 214 Mathematics Building
E-mail: malabika at math dot ubc dot ca
Lectures: Mon,Wed,Fri 9:00 AM to 10:00 AM in Room 460 of Leonard S. Klinck Building.
Office hours: Mon 10-11, Fri 11-12 or by appointment.

TA/Marker: Nicolas Folinsbee
E-mail: nfolinsb at math dot ubc dot ca
Office hours: Fri 12-1 in LSK 300C. Office hours subject to change after the first midterm.

Course information

• This is the course webpage of MATH 321 in Term 2 of the 2018W session (January to April 2019). Here you will find the course outline, suggested homework and practice problems, course policies, exam dates, common handouts and supplementary notes, other course information, and information on available resources.
• First day handout containing syllabus and grading policies.

Piazza links

• Signup link .
• Class link.

Text

UBC course description

• The Riemann or Riemann-Stieltjes integrals
• Sequences and series of functions
• Uniform convergence
• Approximation of continuous functions by polynomials
• Fourier series
• Functions from Rm to Rn
• Inverse and implicit function theorems

Pre-requisite

• MATH 320 or equivalent

Beginning-of-term registration information

• If you are not registered in the course, please do not attend it without the instructor's approval.
• Instructors do not have the authority to "fit you in". Such requests have to be processed by the math department office (Room 121 Mathematics Building) and approved by our undergraduate chair Mark MacLean (maclean at math dot ubc dot ca).

Grading Scheme

• Final Exam 50%
• 2 midterms = 15% + 15% = 30%
• Homework assignments 20%

Exam Dates and Policies

• Midterm info:
• There will be two midterms in MATH 321.
• Midterm 1 on Friday February 1, 9-9:50 AM in class.
• Midterm 1 will cover the material covered in class up to and including January 25. This includes all of Chapter 7 and Sections 6.1-6.6 of Chapter 6.
• Office hours for midterm week: Monday (Jan 28) 10-12, Tuesday (Jan 29) 4-5, Thursday (Jan 31) 10-11. There will be no office hour on Friday.
• Practice problem set . We will go over some of these problems during the in-class review sessions. Hints .
• Please bring your student ID to the exam.
• Midterm 1 .
• Midterm 1 Solutions .
• Midterm 2 on Friday March 15, 9-9:50 AM in class.
• Midterm 2 is non-cumulative, and will cover the material covered in class from February 4 up to and including March 8. This includes all of Chapter 6 and Sections 8.9-8.16 of Chapter 8, plus the additional material we discussed in class.
• Office hours for midterm week: Monday (Mar 11) 10-11:30, Tuesday (Mar 12) 3:45-4:45, Thursday (Mar 14) 10-11. There will be no office hour on Friday.
• Practice problem set . We will go over some of these problems during the in-class review sessions.
• Please bring your student ID to the exam.
• Midterm 2 .
• Midterm 2 Solutions .

• Final exam info:
• The final examination date April 26, 8:30AM-11AM in BUCH A103.
• Office hours during final week:
  • Monday April 22: 9:30-11AM in MATX 1118
  • Tuesday April 23: 2:30-4PM in MATX 1118
  • Thursday April 25: 2:30-4PM in MATX 1118
• The final exam covers the entire course syllabus, as detailed in the week-by-week outline below.
• The final exam will account for 50% of a student's final grade. The remaining 50% will be based on term work. The final exam generally will not be weighted higher for students who perform better on this test than they did during the term, although some allowance may be made for students who perform significantly better (an increase of 30% or more over their term mark). Such decisions are based on the sole discretion of the instructor.
• Past final exams for this course may be found here . Scroll down to the row marked 321.
• Practice problem set .

• Exam aids: No unauthorized electronic devices will be allowed in the midterms or in the final exam. This includes calculators, cell phones, music players and all communication devices. Students should not bring their own formula sheets or other memory aids.

• Please bring your student ID-s to the midterm and the final.

Policy on missed work

• Missed homework: Late submissions will not be accepted, but the two worst homework grades will be dropped.

• Missed midterms: If a student misses a midterm, a documented excuse must be provided or a mark of zero will be entered for that midterm. Examples of valid excuses are an illness which has been documented by a physician and Student Health Services, or an absence to play a varsity sport (your coach will provide you with a letter).

  There will be no make-up midterms. If a valid excuse if provided, the weight of one (and one only!) missed midterm will be transferred to the final examination. To be eligible for this arrangement, you must notify your instructor of your failure to take the test within a week of the missed midterm, and come up with a timeline acceptable to both for producing appropriate documentation for your absence.

  A student may not have both of their midterm weights transferred to the final. If you miss both midterms, you will fail the course regardless of your performance in the final exam.

• Missed final exam: You will need to present your situation to your faculty's Advising Office to be considered for a deferred exam. See the Calendar for detailed regulations . Your performance in a course up to the exam is taken into consideration in granting a deferred exam status (for instance, failing badly normally means you will not be granted a deferred exam). For deferred exams in mathematics, students generally sit the next available exam for the course they are taking, which could be several months after the original exam was scheduled.

Academic misconduct

• UBC takes cheating incidents very seriously. After due investigation, students found guilty of cheating on tests and examinations are usually given a final grade of 0 in the course and suspended from UBC for one year. More information.
• Note that academic misconduct includes misrepresenting a medical excuse or other personal situation for the purposes of postponing an examination or quiz or otherwise obtaining an academic concession.

Homework

• Homework set 1 , due in Canvas at 9AM on Wednesday January 9.
• Homework set 2 , due in Canvas at 9AM on Wednesday January 16.
• Homework set 3 , due in Canvas at 9AM on Wednesday January 23.
• Homework set 4 , due in Canvas at 9AM on Wednesday January 30.
• Homework set 5 , due in Canvas at 9AM on Wednesday February 13.
• Homework set 6 , due in Canvas at 9AM on Wednesday February 20.
• Homework set 7 , due in Canvas at 9AM on Wednesday February 27.
• Homework set 8 , due in Canvas at 9AM on Wednesday March 6.
• Homework set 9 , due in Canvas at 9AM on Wednesday March 13.
• Homework set 10 , due in Canvas at 9AM on Wednesday March 27.
• Homework set 11 , due in Canvas at 9AM on Wednesday April 3.

Practice problems

Here is a list of exercises from the textbook based on the weekly lecture material. They are not meant to be turned in for grading, but it is recommended that you work through them to get a better understanding of the topics covered. Some midterm and final exam problems may be modelled on these exercises. Unlike homework problems, detailed so lutions of these problems will not be posted, but I am happy to include hints upon request.

• Sequences and series of functions
  • Chapter 7: Exercises 1-26. For problem 22, assume that f is Riemann integrable instead of Riemann-Stieltjes inte grable for now.
  • More practice problems on Chapter 7.
• Riemann-Stieltjes integration
  • Chapter 6, Exercises 1-7, 12-15, 17-19.
  • More practice problems on Chapter 6.
• Fourier series
  • Chapter 8, Exercises 12-19.
  • More practice problems will appear here. Stay tuned

Real analysis lecture notes on the web

Here is a list of online lecture notes of similar courses offered at various institutions.

Help outside class

• Each instructor will hold a few (2-3) office hours per week for students in his/her section. See section websites for more details.
• We will be using Piazza (www.piazza.com) as a class discussion platform. This is a question-answer forum designed for getting quick responses from classmates, instructors or TA-s. We encourage you to use this feature extensively and responsibly to pose questions, discuss solutions and ask for clarifications. A sign-up link and a class link will be emailed to you by your instructor. You will need to self-register for this course in Piazza using the link and a valid ubc email address.

Week-by-week course outline

Here is a tentative guideline of the course structure, arranged by week. The textbook pages are mentioned as a reference and as a reading guide. The treatment of these topics in lecture may vary somewhat from that of the text.

• Week 1 (pages 143-154 of the textbook):
  • Pointwise and uniform convergence
  • Definitions and examples
  • Interchanging limits.
  • Applications of pointwise and uniform convergence:
    • Weierstrass M-test
    • A continuous but nowhere differentiable function
    • A space-filling curve
    • Separability of the space of continuous functions on [0,1]

• Week 2 (pages 154-161 of the textbook):
  • Density of polygonal functions in the space C[0,1]
  • A quick review of uniform continuity
  • Bernstein's proof of the Weierstrass approximation theorem
  • Approximation of continuous periodic functions by trigonometric polynomials
  • Algebras and lattices
  • The Stone-Weierstrass theorem - real version

• Week 3 (pages 154-165 of the textbook):
  • Stone-Weierstrass theorem - complex version (a reading exercise)
  • Review of compactness in general metric spaces
  • Equicontinuity
  • Statement of the Arzela-Ascoli theorem

• Week 4 (pages 120-130 of the textbook) :
  • Proof of the Arzela-Ascoli theorem
  • Towards Riemann-Stieltjes integral
  • The Riemann-Stieltjes integral
  • Riemann's condition for Riemann-Stieltjes integrability
  • The space of Riemann-Stieltjes integrable functions

• Week 5 :
  • Midterm review.

• Week 6: (pages 120-135 of the textbook)
  • Riemann-Stieltjes integrability and uniform convergence
  • Functions of bounded variation
  • Jordan's theorem
  • General integrators
  • Integration by parts

• Week 7: (pages 120-135 of the textbook)
  • Integrators of bounded variation
  • Linear functionals on normed linear spaces
  • Riesz representation theorem for continuous linear functionals on C[a,b]

• Week 8 : Winter break
• Week 9 (pages 172-192 of the textbook) :
  • Fourier series
  • Partial Fourier sums as L2 projections
  • Bessel's inequality
  • Plancherel's theorem
  • L2 convergence of Fourier series
  • L2 approximation by continuous periodic functions
  • Dirichlet's formula

• Week 10 (pages 172-192 of the textbook):
  • Properties of the Dirichlet kernel
  • Uniform convergence, or lack thereof, of partial Fourier sums
  • Fejer kernel and Cesaro summability of Fourier series
  • A brief introduction to approximations to the identity

• Week 11:
  • Midterm review

• Week 12 (pages 204-228 of the textbook):
  • The contraction mapping principle
  • Differentiability of multivariate functions
  • Inverse function theorem
  • Implicit function theorem

• Week 13 :
  • Shortcomings of Riemann integration - some workarounds
  • Sets of Lebesgue measure zero
  • Lebesgue's condition on Riemann integrability