MATH 564, Advanced Real Analysis I, Fall 2011

Course web page: http://www.math.mcgill.ca/jakobson/courses/math564.html


Lectures:
  • M, Tu 11:30-12:30, Burnside 1205. Lectures start on September 1.
  • Wed, 12:00-13:00, Burnside 1205.
  • There will be no lectures on monday because of the labor day holiday.
  • Thrusday lectures will be moved to Wednesday, 12:00-13:00, Burnside 1205. The first lecture there will be on Wednesday, september 7.
  • Exceptionally, the lecture on sept. 7 will be repeated from 15:00-16:00 in Burnside 308 (if the room is taken, the lecture will be held in the math lounge on the 12th floor).
  • There will be make-up lectures on Fridays, November 18 and 25, from 11:00-12:00, in Burnside 920.
  • Lecture notes, November 18: page 1, page 2, page 3
  • Lecture notes, November 25: pdf
  • Y. Canzani will replace me on November 29 and 30 at regular times.
  • There will be a review session on Monday, December 12, from 12:00-13:30, in Burnside 1205 (usual room).

  • Instructor: D. Jakobson
    Office: BH1220, Office Hours: Monday, Tuesday, 10:30-11:30; or by appointment
    Tel: 398-3828
    E-mail: jakobson AT math.mcgill.ca
    Web Page: www.math.mcgill.ca/jakobson
    Prerequisite: Math 354, 355 or equivalent
    Marker: Yaiza Canzani
    Email: canzani@math.mcgill.ca

    Text
  • 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).

  • 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.
  • In-class midterm will be held on Tuesday, October 18, 17:30-19:30, in Burnside 920. Solutions: pdf, ps.
  • Take-home midterm: pdf and ps. Due date: Friday, October 21, by 5pm. Solutions to selected problems: pdf and ps
  • There will be a final that will count for 50% (or 80%) of your grade
  • Date and time: December 15, 9:00-12:00
  • Practice problems: pdf

  • Assignments

  • Problem Set 1 (due Wednesday, September 28): ps and pdf.
  • Problem Set 1, part 2 (due Wednesday, September 28): ps and pdf.
  • Solutions (jpeg): page 1, page 2, page 3, page 4, page 5, page 6, page 7, page 8, page 9, page 10, page 11,
  • Problem Set 2 (due Wednesday, October 12): ps and pdf.
  • Solutions (jpeg): 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
  • Problem Set 2, part 2 (due Wednesday, October 12): ps and pdf. Solutions, part 1: pdf and ps
  • Problem Set 3 (due Monday, November 7): ps and pdf.
  • Solutions (jpeg): 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
  • Problem Set 4 (due Monday, November 21): ps and pdf. Solutions: pdf
  • Problem Set 5 (due Tuesday, December 6): ps and pdf. Part 2: ps and pdf. Solutions: pdf

  • Handouts
  • Around Fubini's theorem: "counterexamples": J. Feldman, J.M. Boardman; papers by A.V. Uglanov and Mattner. A paper of E. Hewitt: Integration by parts for Stieltjes integrals
  • Karl Stromberg's paper about Banach-Tarski paradox.
  • Fubini's theorem, summary of the material from Rudin: pdf and ps
  • Gauss-Bonnet theorem: lecture notes by Walkden; wikipedia entry.
  • Handout on Bernstein approximation theorem: 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.
    Grading:
  • Your final mark will be the largest of the following: [20% Assignments + 30% Midterm + 50% Final]; OR [20% Assignments + 80% 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.
    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.

  • Courses next semester
  • R. Yassawi. Math 740: Measurable and symbolic dynamical systems
  • 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: helpdesk@math.mcgill.ca
    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.