MAT C37 Introduction to Real Analysis (Winter 2018)

Instructor: Prof. Robert Haslhofer

Contact Information: roberth(at)math(dot)toronto(dot)edu, IC477

Website: http://www.math.toronto.edu/roberth/C37.html

Lectures: Tuesday 9--11 and Thursday 10--11 in IC326

Office Hours: Tuesday 11--12 and 16--17 in IC477

Textbook: E. Stein, R. Shakarchi: Real Analysis -- Measure theory, Integration and Hilbert Spaces, Princeton University Press

Secondary Reference: H. Royden, P. Fitzpatrick: Real Analysis, Pearson

Topics to be covered: Analysis in R^d, Lebesgue measure, Lebesgue integration, Differentiation and Integration, L^p spaces, Hausdorff measure, Fourier analysis

Grading Scheme: Homework 30%, Midterm 30%, Final Exam 40%

Midterm Exam: Tuesday Feb 13th from 9--11 in IC326

Final Exam: Saturday April 21st from 19--22 in IC120

Remarks: Please discuss lectures and homework problems among yourselves and with me, and consult other sources. But write up your assignments in your own words, and be ready to defend them! Your work will be judged on the clarity of your presentation as well as correctness and completeness.


Homework problem sets:
HW1: Problem Set 1 HW2: Problem Set 2 Practice Problems: Practice Problems HW3: Problem Set 3 HW4: Problem Set 4 HW5: Problem Set 5 Practice Problems for Final: Practice Problems for Final


Weekly schedule:

Week 1 (Jan 8-14)
Tuesday: Why do we need measure and integration theory?
Thursday: Review of basic properties of Euclidean space R^d

Week 2 (Jan 15-21)
Tuesday: Review of basic notions from topology
Thursday: Rectangles and cubes

Week 3 (Jan 22-28)
Tuesday: The exterior measure
Thursday: The exterior measure (continued)

Week 4 (Jan 29-Feb 4)
Tuesday: Measurable sets
Thursday: The Lebesgue measure

Week 5 (Feb 5-11)
Tuesday: The Lebesgue measure (continued)
Thursday: The Cantor-Lebesgue function

Week 6 (Feb 12-18)
Tuesday: Term Test
Thursday: Measurable functions

Week 7 (Feb 26-Mar 4)
Tuesday: Approximation by simple functions, Littlewood's three principles
Thursday: Integration of simple functions

Week 8 (Mar 5- 11)
Tuesday: Integration of bounded functions, comparison with the Riemann integral
Thursday: Integration of nonnegative and integrable functions

Week 9 (Mar 12-18)
Tuesday: Fatou's lemma, monotone convergence theorem
Thursday: Dominated convergence theorem

Week 10 (Mar 19-25)
Tuesday: The space L^1 of integrable functions, completeness
Thursday: dense subsets

Week 11 (Mar 26 - Apr 1)
Tuesday: The Hilbert space L^2
Thursday: Fourier series

Week 12 (Apr 2 - 8)
Tuesday: Lebesgue differentiation theorem
Thursday: Fundamental theorem of Calculus revisited