Math 321 - Real Variables II - Spring 2012

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 104 Mathematics Building.
Office hours: Mon 10-11 am, Wed 11 am-12 noon or by appointment.

Course information

• Course information handout containing syllabus and grading policies.

Week-by-week course outline

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

• Week 1 (pages 143-146 of the textbook):
  • Pointwise and uniform convergence
  • Definitions and examples.
• Week 2 (pages 149-154 of the textbook):
  • Interchanging limits
  • Some applications
• Week 3 (pages 147-148 of the textbook):
  • The space of bounded functions
  • Weierstrass M-test
  • The Weierstrass approximation theorem
• Week 4 (pages 159-161 of the textbook):
  • Bernstein's proof of Weierstrass approximation theorem
  • Approximation of continuous periodic functions by trigonometric polynomials
• Week 5 (pages 154-158 of the textbook):
  • Equicontinuity
  • Arzela-Ascoli theorem
• Week 6 (pages 161-165 of the textbook):
  • Algebras and lattices
  • The Stone-Weierstrass theorem
• Week 7 :
  • Midterm review.
• Week 8 :
  • Winter break.
• Week 9 (see the textbooks in real analysis by T. Hildebrandt, or N. Carothers) :
  • Towards Riemann-Stieltjes integral
  • Functions of bounded variation
  • Jordan's theorem
• Week 10 (pages 120-130 of the textbook):
  • The Riemann-Stieltjes integral
  • Riemann's condition for Riemann-Stiletjes integrability
  • An integral formula for total variation
• Week 11 :
  • The space of Riemann-Stieltjes integrable functions
  • Closure under uniform convergence
  • General integrators
• Week 12:
  • Integration by parts
  • Integrators of bounded variation
  • Riesz representation theorem
• Week 13:
  • Fourier series
  • L^2 convergence of Fourier series
  • Bessel's inequality
  • Plancherel's theorem

Homework

• Homework 1 due on Friday January 13.
• Homework 2 due on Friday January 20.
• Homework 3 due on Friday January 27.
• Homework 4 due on Friday February 3.
• Homework 5 due on Friday February 10.
• Homework 6 due on Friday February 17.
• Homework 7 due on Friday March 9.
• Homework 8 due on Friday March 16.
• Homework 9 due on Friday March 23.
• Homework 10 due on Friday March 30.
• Homework 11 not to be turned in.

Midterm

• Practice problem set for midterm 1.