MATH 565, Advanced Real Analysis II, Winter 2012

Course web page:

  • MWF 11:30-12:30, Burnside 920. Lectures start on January 9.

  • Instructor: D. Jakobson
    Office: BH1220, Office Hours: Monday, 10:30-11:30; Wednesday, 12:30-13:30; or by appointment
    Tel: 398-3828
    E-mail: jakobson AT
    Web Page:
    Prerequisite: Math 564
    Marker: Mark Bumagin

  • Required: G. Folland. Real analysis. Modern techniques and their applications. Second edition. Pure and Applied Mathematics, Wiley-Interscience Publication, New York, 1999.
  • Recommended: W. Rudin, Real and Complex Analysis, McGraw-Hill (on 3-hour reserve at Schulich library); G. Folland, Real Analysis, Modern techniques and Their Applications (on 3-hour reserve at Schulich library).

  • Syllabus:
  • Continuation of Math 564. Review of theory of measure and integration; product measures, Fubini's theorem; Lp spaces, basic principles of Banach spaces; Riesz representation theorem for C(X); Hilbert spaces; Fourier analysis; further material as time permits.
  • Course summary

  • Tests
  • There will be a midterm that will count 30% of the grade. You can write a take-home midterm, in-class midterm or both. Your grade will be the maximum of the 2 scores.
  • Take-home midterm will be distributed on Friday, February 17; it will be due back on Monday, February 27.
    Midterm (pdf). Solutions: solution 1, solution 2
  • In-class midterm will be held on Friday, March 2, 11:00 - 12:30, Burnside 1B24. Solutions (pdf).
  • There will be an in-class final; its weight under various marking options is deccribed in the Grading section below.
  • Date: Friday, April 20, 14:00-17:00, Room TBA

  • Oral Presentation
    Students will have an option of giving a 30-minute oral presentation in class on a topic chosen by the student in consultation with the instructor. If a student chooses that option, the presentation will be evaluated, and will count for 10% of the grade.
    List of possible topics:
  • Rearrangement ineqaultities
  • Steiner symmetrization
  • Sobolev inequalities
  • Poincare's inequalities
  • Banach-Tarski paradox
  • Ruziewicz problem
  • Lie groups, Haar measure
  • Ergodic theorem
  • Isoperimetric inequalities
  • Rayleigh quotient, eigenvalues
  • Co-area formula
  • Cheeger constant
  • Infinite-dimensional spaces
  • Other topics to be announced

  • Grading:
  • Your final mark will be the largest of the following: [20% Assignments + 30% Midterm + 50% Final]; OR [20% Assignments + 80% Final]; OR [20% Assignments + 30% Midterm + 10% Presentation + 40% Final]; OR [20% Assignments + 10% Presentation + 70% Final].
  • In the event of extraordinary circumstances beyond the University's control, the content and/or evaluation scheme in this course is subject to change.

  • WebCT: Your scores on assignments, presentation, midterm, final, and your final mark will be posted on WebCT
    Supplemental: There will be a supplemental exam, counting for 100% of the supplemental grade. No additional work will be accepted for D, F or J.
    Homework will be assigned in class and will be due by 5pm by the specified deadline.


  • Problem Set 1, due Monday, January 30: pdf
  • Problem Set 1, part 2, due Monday, January 30: pdf
  • Assignment 1, solutions: page 1, page 2, page 3, page 4, page 5, page 6, page 7, page 8, page 9, page 10, page 11, page 12, page 13, page 14, page 15, page 16, page 17, page 18, page 19, page 20, page 21, page 22, page 23, page 24, page 25, page 26, page 27, page 28, page 29, page 30, page 31, page 32, page 33, page 34, page 35, page 36, page 37, page 38, page 39, page 40, page 41, page 42, page 43, page 44
  • Problem Set 2, due (tentatively) Monday, February 13: pdf. Solutions: part 1, part 2, part 3
  • Problem Set 2, part 2, due (tentatively) Monday, February 13: pdf
  • Problem Set 3, due Monday, March 19: pdf. Solutions: solution 1, solution 2, solution 3
  • Problem Set 4, due Wednesday, april 11: pdf. Exercises 42, 44, 46 are extra credit. Solutions: solution 1, solution 2.

  • Course material from previous courses at McGill:
  • Old Math 564, Fall 2008 and Math 564, Fall 2009 web pages, D. Jakobson. Old Math 564, Fall 2010 web page, V. Jaksic.
  • Old Math 565, Winter 2009 and Math 565, Winter 2010 web pages, D. Jakobson. Old Math 565, Winter 2011 web page, V. Jaksic.
  • Old Math 354 and Math 355 web pages, D. Jakobson
  • Sam Drury's lecture notes for MATH 354 and MATH 355
  • Old Math 366 web page, D. Jakobson, Fall 2007
  • Vojkan Jaksic's Lecture Notes in Spectral Theory, ps and pdf.

  • Lecture notes in Measure Theory
  • S. Sternberg's Theory of functions of a real variable lecture notes.

  • Web links in Analysis
  • Metric space, Topology glossary, Functional analysis in Wikipedia
  • Norm, Holder's inequality, Minkowski inequality, Lp space, Hilbert space, Banach space, Cantor set, p-adic numbers in Wikipedia
  • Notes on differentiation of functions of several variables, implicit function theorem
  • Companion notes to Rudin's (undergraduate!) book
  • Harmonic Analysis page by Terry Tao (there is a lot of advanced stuff there)

  • HELPDESK and their email:
    NOTICE: McGill University values academic integrity. Therefore, all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the Code of Student Conduct and Disciplinary Procedures (see McGill web page on Academic Integrity for more information).
    NOTICE: In accord with McGill University's Charter of Student Rights, students in this course have the right to submit in English or in French any work that is to be graded.
    NOTICE: In the event of extraordinary circumstances beyond the University's control, the content and/or evaluation scheme in this course is subject to change