MATH 354: Honors Analysis 3

Fall 2012

Course web page:

  • MWF 13:30-14:30, Burnside 1B23
  • The lectures will start on Wednesday, September 5.
  • The last lecture is on December 5.
  • There will be office hours on December 11 (Tuesday) and December 12 (Wednesday), from 12:00-14:00.

  • Instructor: D. Jakobson
    Office: BH1220
    Office Hours: Wednesday, 11:30-13:30; or by appointment
    Tel: 398-3828
    E-mail: jakobson AT
    Web Page:
    Review sessions:
  • Discussion leader: Michael Snarski
  • Novermber 15, 19:00-21:00, Stewart Biology Building, S1/3.
  • Novermber 22, 18:30-20:30, Burnside 1205
  • Novermber 29, 18:00-20:00, Leacock Building, LEA 219
  • Michael wrote some notes for the course

  • Markers:
  • Marc-Adrien Mandich
  • Janine Bachrachas, Burnside 1029.

  • Decription of BSc-MSc program

    Prerequisites: Math 255 or equivalent
  • Real Analysis: Measure theory, integration and Hilbert Spaces, by E. Stein and R. Shakarchi, Princeton Lectures in Analysis 3, Princeton University Press.
  • 2010 Lecture notes taken by Robert Gibson.
  • Notes on Introductory Point-Set Topology by Allen Hatcher.

  • Syllabus: Introduction to metric spaces. Completenes, Compactness, Connectedness. Measure theory and Integration. Implicit and inverse function theorems.
    Assignments: There will be be several assignments. Due dates will be announced in class and on the course web page. Late assignments will not be accepted, except in cases of emergency. Depending on availability of TA and other factors, some problems may not be marked.
  • Assignment 1, due Monday, September 24: pdf. Problems 4, 8 and 9 are extra credit. Solutions (selected problems) pdf. Solutions (complete) pdf
  • Assignment 2, due Monday, October 1: pdf and latex. Problems 1b, 6, 8, 9, 10 are extra credit. Solutions (selected problems) pdf. Solutions (complete) pdf.
  • Assignment 3, due Monday, October 15 (postponed to Wednesday, October 17): pdf and latex. Problems 1iii, 8, 9, are extra credit. Solutions: pdf. Solutions, improved: pdf.
  • Assignment 4, due Friday, October 19 (postponed to Monday, October 22): pdf. Problems 1 and 2e are extra credit. Solutions (problems 1, 2, 3): pdf. All problems: pdf.
  • Assignment 5, due Monday, November 19: pdf. Problems in latex: latex and pdf. Solutions: version 1, version 2, version 3, version 4.
  • Practice problems with solutions (not for credit): pdf.
  • Assignment 6, due Monday, December 3: pdf. Problem 22 is extra credit. Solutions: version 1, version 2, version 3, version 4, version 5.

  • Handouts:
  • Elementary proof of Tychonoff's theorem via nets Paul Chernoff, American Math. Monthly, 99 (1992), pp. 932-934.
  • Differentiation in Function spaces: an example: ps and pdf
  • Handout on Bernstein approximation theorem: ps and pdf
  • Handout on Stone-Weierstrass theorem: ps and pdf
  • Handout on miscellaneous properties of metric spaces: ps and pdf
  • Handout on Baire's Category theorem and Uniform Boundedness Principle ps and pdf
  • Handout on the Intermediate Value theorem ps and pdf
  • Handout on Inverse Function and Implicit Function theorems in R^n ps and pdf
  • Summary of course material in 2006 course Math 354: ps and pdf
  • Summary of the course material in the Fall 2010, compiled by Robert Gibson
  • Summary from Stein/Shakarchi, chapters 1 and 2
  • Summary of the rest of the material in the course

  • Midterm:
  • There will be a choice of an in-class or take-home midterm. Both midterms would be marked. You can attempt both tests, your mark will be the highest of the 2 marks that you receive.
  • In-class midterm: Wednesday, October 31, Rutherford Physics building, Room 112, 17:45-19:45.
  • Missed midterm cannot be redone. If you miss the midterm for any reason, the weights for your mark will be: Assignments 20%, Final 80%.
  • In-class midterm: pdf
  • Take-home midterm: pdf Due Wednesday, November 7. Do any SIX problems.
  • Home midterm solutions: pdf. Some problems also appeared in one of the assignments in the graduate course that I taught previously: problem set and solutions; the file is fairly long, the relevant solutions begin on page 21. Solutions to the problem about the Hausdorff distance can be found here.
  • In 2010, Robert Gibson compiled some definitions and theorems discussed in Math 354 in 2010 before the midterm.

  • Final:
  • There will be a three hour final exam. Date/Time: December 14, 14:00-17:00, Burnside 1B23 and 1B45
  • 2010 Final: pdf. Note that the material for the problems 3,5,6 was not covered in 2012 and will not appear on this year's final.
  • SUMS website should have arxived exams.
  • Supplemental: There will be a supplemental exam, counting 100% of the supplemental grade. No additional work will be accepted for D, F, or J.
  • Namdar Homayounfar compiled these notes for the course, this is a very good summary.

  • Grading: Your final mark will be the largest of the following: [20% Assignments + 30% Midterm + 50% Final]; OR [20% Assignments + 80% Final].
    WebCT: Your scores on assignments, midterm, final, and your final mark will be posted on WebCT
    Course material from previous courses at McGill:
  • Prof. V. Jaksic: 2009 Math 354
  • A. Tomberg took Lecture notes of Prof. Jaksic's lectures.
  • Prof. D. Jakobson: 2010 Math 354
  • 2010 Lecture notes taken by Robert Gibson.
  • Prof. D. Jakobson: 2006 Math 354
  • Linear algebra review (D. Jakobson): A note about determinants, ps and pdf.
  • Material from old Math 265 Course Pak (Prepared by Taylor and Labute): Implicit Function Theorem pdf and ps
  • Sam Drury's lecture notes for MATH 354 and MATH 355

  • Web links in Geometry and Topology
  • Lecture notes by Allen Hatcher
  • Lecture notes in general topology by Jan Derezinski
  • Glossary (wikipedia)
  • Another glossary
  • Rough guide to point-set topology
  • A wikibooks course in topology
  • A small handout of topological terms, prepared by P. Rosenthal
  • Introduction to Hausdorff distance: paper by J. Henrikson; page at Wapedia; applications to image recognition: Hausdorff distance between convex polygons, N. Gregoire and M. Bouillot.

  • 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 book
  • Yahoo group in Harmonic Analysis + a page with listings of conferences, successors to Terry Tao's old page on Harmonic analysis.
  • Terry Tao's blog
  • HELPDESK and their email:
    WEB LINKS in Calculus, Algebra, Geometry and Differential Equations.
    Courses next semester that you may find interesting:
  • Math 355, Honors Analysis 4 (J. Toth)
  • Hyperbolic Geometry and Automorphic Forms Math 480/Math 693/ Math 740 (D. Jakobson and J. Toth)
  • A reading course on groups and expanders (L. Addario-Berry, D. Jakobson, M. Pichot)
  • Topics in Probability (Brownian Motion) Math 784 (L. Addario-Berry)

  • 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