MATH 565, Advanced Real Analysis II, Winter 2017

Course web page:

  • Monday, Tuesday, Thursday 9:30-10:30, Burnside 1205. Lectures start on Thursday, January 5 and end on Tuesday, April 11.

  • Instructor: D. Jakobson
    Office: Burnside Hall 1220, Office Hours: Monday and Thursday, 10:30-11:30 (after class), or by appointment.
    Tel: 398-3828
    E-mail: jakobson AT
    Web Page:
    Prerequisite: Math 564
    Marker: Tyler Cassidy

  • 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, 3rd edition. McGraw-Hill Book Co., New York, 1987

  • Syllabus:
  • Continuation of Math 564. We shall cover remaining material from Chapters 6-8 in Folland: Lp spaces and interpolation, selected topics from Chapter 7 (Radon measures), Fourier Analysis. Then we shall discuss selected topics from Chapters 9-11 in Folland, as time permits (to be chosen from: distributions, Sobolev spaces, topics in Probability, Haar measure and Hausdorff measure), and possibly other topics.

  • 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.
  • The home midterm will be posted on Friday, February 24, and will be due on Monday, March 6, in class.
  • The class midterm will be held on Tuesday, March 7, from 5:45-7:45pm, in Burnside 708.
  • Take home midterm, due Monday, March 6.
  • Take home final: pdf. It is due on Monday, April 24.
  • 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 Wednesday, April 26, from 14:00-17:00, in Burnside 1B23.

  • 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
  • Fractal sets
  • PDE
  • Spectral theory

  • 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 Tuesday, January 24 (extended): Assignment 1
  • Hausdorff measure problems, not for credit.
  • Assignment 2, due Monday, Febuary 13 (extended): Assignment 2
  • There is a typo in Problem 45, page 208. Please, see the correction here
  • Assignment 3, due Thursday, Febuary 23 (extended): Assignment 3
  • Assignment 4, due date to be announced: Assignment 4
  • Assignment 5, due date to be announced: Assignment 5

  • Lecture notes on related topics

  • Bernstein Approximation Theorem (from Sam Drury's notes)
  • 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. Math 564, Fall 2016 web page, Jeff Galkowski.
  • Old Math 565, Winter 2009, and Math 565, Winter 2010, Math 565, Winter 2012 and Math 565, Winter 2016 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