MAT C37 Introduction to Real Analysis (Winter 2019)

Instructor: Prof. Robert Haslhofer

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


Lectures: Tuesday 9--11 in IC 326 and Thursday 10--11 in IC320

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

PREREQUISITE: MATB43. Note: If you have not taken MATB43 then you can NOT take MATC37 (The only case where I consider waiving this is if you are exceptionally strong, e.g. CGPA above 3.5)

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

Midterm Exam: Tuesday Feb 26 from 9--11 in IC326

Final Exam: Monday April 15 from 19--22 in IC200

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 7-13)
Tuesday: Why do we need measure and integration theory?
Thursday: Review of basic properties of Euclidean space R^d

Week 2 (Jan 14-20)
Tuesday: Review of basic notions from topology
Thursday: Rectangles and cubes

Week 3 (Jan 21-27)
Tuesday: The exterior measure
Thursday: The exterior measure (continued)

Week 4 (Jan 28-Feb 3)
Tuesday: Snowstorm
Thursday: Measurable sets

Week 5 (Feb 4-10)
Tuesday: The Lebesgue measure
Thursday: The Lebesgue measure (continued)

Week 6 (Feb 11-17)
Tuesday: Snowstorm
Thursday: Measurable functions

Week 7 (Feb 25-Mar 3)
Tuesday: Term Test
Thursday: Approximation by simple functions, Integration of simple functions

Week 8 (Mar 4- 10)
Tuesday: Integration of bounded functions, comparison with the Riemann integral
Thursday: Integration of nonnegative and integrable functions

Week 9 (Mar 11-17)
Tuesday: Fatou's lemma, monotone convergence theorem
Thursday: Dominated convergence theorem

Week 10 (Mar 18-24)
Tuesday: The space L^1 of integrable functions, completeness
Thursday: dense subsets

Week 11 (Mar 25 - 31)
Tuesday: The Hilbert space L^2
Thursday: Fourier series

Week 12 (Apr 1 - 7)
Tuesday: Lebesgue differentiation theorem
Thursday: Fundamental theorem of Calculus revisited