MATH 565, Advanced Real Analysis II, Winter 2016

Course web page:

  • Monday, Tuesday, Thursday 11:30-12:30, Burnside 1205. Lectures start on January 7.

  • Instructor: D. Jakobson
    Office: BH1220, Office Hours: Tuesday, Thursday, 12:30-13:30.
    Tel: 398-3828
    E-mail: jakobson AT
    Web Page:
    Prerequisite: Math 564
    Marker: Mikhail Karpukhin

  • Required: W. Rudin, Real and Complex Analysis, 3rd edition. McGraw-Hill Book Co., New York, 1987
  • Recommended: G. Folland. Real analysis. Modern techniques and their applications. Second edition. Pure and Applied Mathematics, Wiley-Interscience Publication, New York, 1999.

  • Syllabus:
  • Continuation of Math 564. We shall cover (approximately) the material from Chapters 5-9 in Rudin: Complex measures, absolute continuity, Riesz representation theorem, differentiation of measures, Lp spaces, product measures, Fubini's theorem, Fourier transforms, further material as time permits.

  • 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, due Monday, March 7.
  • There will be a final exam; its weight under various marking options is described in the Grading section below.
  • You can write an in-class final, a take-home final, or both. Your grade will be the maximum of the 2 scores.
  • The in-class final exam will be held on Thursday, April 28, from 14:00-17:00, room to be announced.
  • Things to review for the final.

  • 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 20% of the grade.
    List of possible topics (work in progress):
  • 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
  • Poisson Integral
  • Other topics to be announced

  • Grading:
  • Your final mark will be the largest of the following: [15% Assignments + 30% Midterm + 55% Final]; OR [15% Assignments + 85% Final]; OR [15% Assignments + 20% Midterm + 20% Presentation + 45% Final]; OR [15% Assignments + 20% Presentation + 65% 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.

  • MyCourses: Your scores on assignments, presentation, midterm, final, and your final mark will be posted on MyCourses
    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.


  • Assignment 1, due Thursday, January 28: Rudin, Chapter 4, Problems 7, 11, 13, 18 (extra credit); Chapter 5, Problems 3, 5, 7. Problems 8, 9 moved to Assignment 2.
  • Assignment 1, Part 2 (extra credit): pdf, due date to be announced.
  • Assignment 2, due Monday, February 8. Do any 7 problems. Rudin, Chapter 5, Problems 8, 9, 11, 14, 15, 16, 17, 21, 22.
  • Assignment 2, Part 2 (extra credit): pdf, due date to be announced.
  • Assignment 2, Part 3 (extra credit): pdf, due date to be announced.
  • Assignment 3, due February 22: Rudin, Chapter 6, Problems 4, 6, 7, 9, 10 (extra credit), 11, 13 (extra credit).
  • Assignment 4, due Monday, March 21. Rudin, Chapter 7, Problems 2 (extra credit), 5, 7 (extra credit), 10, 11, 12, 13, 15 (extra credit), 18 (extra credit), 22 (extra credit).
  • Assignment 5, due Monday, March 28. Do any 5 problems. Rudin, Chapter 8, Problems 2, 4, 5 (extra credit, 10 points), 8 (extra credit), 12, 13 (extra credit), 14, 16.
  • Assignment 6, due Monday, April 11. Do any 6 problems. Rudin, Chapter 9, Problems 2, 3, 6, 9 (extra credit), 11, 12, 13, 15, 18, 19.

  • Handouts

  • A. Hatcher Notes on Introductory Point-Set Topology
  • V. Jaksic. Topics in Spectral theory (including complex measures, Fourier transform of measures, Poisson transform, spectral theory of self-adjoint operators, spectral theory of rank one perturbations)
  • A short course on rearrangement inequalities, written by Prof. Almut Burchard at the University of Toronto [from Math 564]
  • Steiner symmetrization slides by Andrejs Treibergs at the University of Utah [from math 564].
  • A 2003 paper by V. Milman and B. Klartag about Minkowski symmetrization [from Math 564]
  • A 2002 paper by B. Klartag, titled: 5n Minkowski Symmetrizations Suffice to Arrive at an Approximate Euclidean Ball; its arxiv version [from Math 564]
  • A 2004 paper by B. Klartag about the rate of convergence of sequences of Steiner and Minkowski symmetrizations; its arxiv version [from Math 564]
  • A 1990 paper by E. Lieb, containing the proof of Hausdorff-Young inequality with sharp constants and characterizing the minimizers.

  • 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 and Math 565, Winter 2012 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
  • 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