MATH 565, Advanced Real Analysis II, Winter 2009

Course web page:

  • MWF 9:30-10:30, Burnside 1120. Lectures start on January 9.
  • The instructor will be out of town until january 8; the first lecture will be given on january 9, there will be two make-up lectures in the following weeks.
  • Make-up lectures: Wednesday, January 21 and 28, 13:30-14:30, Burnside 1234.
  • No class Friday, April 10.
  • Tuesday, April 14: lecture cancelled.
  • Office hours during the week of April 13-17:
    Tuesday, April 14, 16:15-17:15
    Thursday, April 16, 16:00-17:00
    Friday, April 17, 12:00-13:00

  • Instructor: D. Jakobson
    Office: BH1212, Office Hours: M 11:00-12:00, W 10:30-11:30 (may be changed), or by appointment
    Tel: 398-3828
    E-mail: jakobson AT
    Web Page:
    Prerequisite: Math 564
    Marker: Phil Sosoe

  • Required: W. Rudin, Real and Complex Analysis, McGraw-Hill.

  • Syllabus: Continuation of Math 564; Chapters 7, 8, 9, 11 in Rudin; further material as time permits (to be announced later).
  • There will be a midterm.
  • 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: pdf and ps
  • In-class midterm: Wednesday, March 11, 17:00-18:30, Burnside 1234.
  • You can write a take-home final, in-class final or both. Your grade will be the maximum of the 2 scores.
  • The in-class final exam will be held on April 17, 14:00-17:00, in Leacock 14.
  • The take-home final exam will be given out on April 17 and due back on April 24. Take-home final: pdf and ps
  • List of things to review for the final and the quals: pdf and ps.

  • 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 15% of the grade.
    List of topics:
  • Banach-Tarski paradox
  • Ruziewicz problem
  • Lie groups, Haar measure
  • Lie group actions, measures on the sphere at infinity (could be challenging)
  • Ergodic theorem
  • Isoperimetric inequalities
  • Rayleigh quotient, eigenvalues
  • Co-area formula
  • Cheeger constant
  • Infinite-dimensional spaces
  • Topics in harmonic analysis [expanded list to be supplied]

  • Grading: Your final mark will be the largest of the following: [25% Assignments + 30% Midterm + 45% Final]; OR [25% Assignments + 75% Final]; OR [15% Presentation + 25% Assignments + 30% Midterm + 30% Final]; OR [15% Presentation + 25% Assignments + 60% Final]
    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 date to be announced: ps and pdf.
    Solutions: pdf
  • Hausdorff measure problem set. Part 1: ps and pdf.
  • Problem Set 2, due date to be announced: ps and pdf. Solutions to selected problems: pdf
  • Problem Set 3, due date to be announced: ps and pdf
  • Problem Set 4, due date to be announced: ps and pdf

  • Course material from previous courses at McGill:
  • Web page for Math 564, D. Jakobson, Fall 2008.
  • 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
  • Vojkan Jaksic's Lecture Notes in Spectral Theory, ps and pdf.

  • 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).