MATH 564, Advanced Real Analysis I, Fall 2009

Course web page:

  • MWF 9:30-10:30, Burnside 1205. Lectures start on September 4.
  • Extra lecture: Monday, September 14, 11:30-12:30, Burnside 1234.
  • There will be a lecture on December 3 (Monday schedule), usual time and place.

  • Instructor: D. Jakobson
    Office: BH1212, Office Hours: Monday, 10:30-11:30; Wednesday, 11:30-12:30; or by appointment
    Tel: 398-3828
    E-mail: jakobson AT
    Web Page:
    Prerequisite: Math 354, 355 or equivalent
    Marker: Tayeb Aissiou

  • Required: E. Lieb and M. Loss. Analysis. 2nd edition. Graduate Studies in Mathematics, 14. AMS, 2001.
    The book is on 3-hour reserve at Rosenthal Library (Burnside, 11th floor).
  • 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).

  • Handouts
  • Fubini's theorem, summary of the material from Rudin: pdf and ps
  • A short course on rearrangement inequalities, written by Prof. Almut Burchard at the University of Toronto.
  • Steiner symmetrization slides by Andrejs Treibergs at the University of Utah.
  • A 2003 paper by V. Milman and B. Klartag about Minkowski symmetrization.
  • A 2002 paper by B. Klartag, titled: 5n Minkowski Symmetrizations Suffice to Arrive at an Approximate Euclidean Ball; its arxiv version.
  • A 2004 paper by B. Klartag about the rate of convergence of sequences of Steiner and Minkowski symmetrizations; its arxiv version.

  • Syllabus: 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; further material as time permits.
  • 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. The midterm will cover the material in Lieb and Loss, Chapter 1; Chapter 2, sections 2.1 - 2.4, 2.7-2.13.
  • Take-home midterm: pdf and ps. Due date: Wednesday, November 11, by 3pm.
  • In-class midterm will be held at the end on Wednesday, November 4, between 18:00-19:30pm, in Room 1205.
  • In-class final: Monday, December 7, 9:00-12:00, room RUTHERFORD PHYSICS, room 114, see exam schedules
  • Take-home final: pdf and ps. It is due on December 15 (the due date extended by one day).

  • Grading:
  • Your final mark will be the largest of the following: [30% Assignments + 30% Midterm + 40% Final]; OR [30% Assignments + 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, 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 Wednesday, October 7): ps and pdf.
  • Hausdorff measure problem set 1 (optional, due date to be announced): ps and pdf.
  • Problem Set 2 (due Friday, October 9): ps and pdf. Part 2: ps and pdf.
  • Problem Set 3 (due Friday, October 23): ps and pdf.
  • Problem Set 4 (due Friday, October 30): ps and pdf.
  • Problem Set 5 (due Wednesday, December 2): ps and pdf.

  • Course material from previous courses at McGill:
  • Old Math 564, Fall 2008 web page, D. Jakobson.
  • Old Math 565, Winter 2009 web page, D. Jakobson.
  • Old Math 354 web page, D. Jakobson, Fall 2006
  • Sam Drury's lecture notes for MATH 354 and MATH 355
  • Old Math 366 web page, D. Jakobson, Fall 2007

  • Web links in Analysis
  • Construction of Lebesgue measure: Facenda Aguirre and Freniche; R. Simha; Erik van Erp; Christopher E. Heil.
  • 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.