*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 handout containing syllabus and grading policies.

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 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.

- Practice problem set for midterm 1.