MATH 354: Honors Analysis 3

Fall 2010

Course web page:

  • MWF 11:30-12:30, Burnside 1B23
  • The lectures will start on September 8.
  • There will be a make-up lecture on wednesday, september 15, from 18:00-20:00, in Burnside 920.
  • There will be no lecture on Friday, december 3.
  • There will be review sessions on Tuesday, December 7, from 12:30-13:30, room Burnside 1214.
  • There will be review sessions on Thursday, December 9, from 17:30-19:00, room Burnside 920.
  • Lecture notes taken by Robert Gibson.

  • Instructor: D. Jakobson
    Office: BH1220
    Office Hours: Wednesday, 10:30-11:30 and 12:30-13:30; or by appointment
    Tel: 398-3828
    E-mail: jakobson AT
    Web Page:
    Markers: Daniel Bernucci

    Decription of BSc-MSc program

    Prerequisites: Math 255 or equivalent
    Text: Elements of the Theory of Functions and Functional Analysis, by A. Kolmogorov and S. Fomin, Dover Publications.
    Syllabus: Introduction to metric spaces. Completenes, Compactness, Connectedness. Multivariable differential calculus, 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 27: ps and pdf. Problems 7 and 8 are extra credit. Solutions to selected problems: ps and pdf. Additional solutions: ps and pdf.
  • Assignment 2, due Wednesday, October 13 (extended): ps and pdf. Problem 4 is extra credit. In problem 10 ii), there should be an open interval (a,b), endpoints not included. Old version had a typo, it was corrected in the new version. Solutions to selected problems: ps and pdf. Part 2: ps and pdf. Part 3: ps and pdf.
  • Assignment 3, due Friday, October 22. ps and pdf. Problems 1 and 5 will be due after the midterm. Solutions to selected problems, part 1: ps and pdf. Part 2: ps and pdf.
  • Assignment 4, due Wednesday, November 17. ps and pdf. Solutions: ps and pdf.
  • Practice problems (not for credit): ps and pdf. Solutions: ps and pdf.
  • Assignment 5, Wednesday, November 24. ps and pdf. Problems 1 and 2 are extra credit. Solutions: ps and pdf.
  • Assignment 6, due Wednesday, December 1. ps and pdf. Solutions: ps and pdf.
  • Practice problems II (not for credit), with solutions: ps and pdf

  • 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 this year, compiled by Robert Gibson

  • 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 27, Stewart Biology S3/3, 18:00-20:00. ps and pdf
  • Take-home midterm: pdf and ps. Due Monday, November 8. Solution to Problem 3: pdf and ps.
  • Missed midterm cannot be redone. If you miss the midterm for any reason, the weights for your mark will be: Assignments 20%, Final 80%.
  • Solution to the practice problem discussed on October 25: ps and pdf.
  • Robert Gibson compiled some definitions and theorems discussed in the course so far.

  • Final: There will be a three hour final exam. Tentative Date/Time: December 10, 14:00-17:00, Leacock 109 and 110.

    Supplemental: There will be a supplemental exam, counting 100% of the supplemental grade. No additional work will be accepted for D, F, or J.
    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: 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 (D. Jakobson)
  • Student Seminar in Mathematical Physics 2011 (V. Jaksic and R. Seiringer)
  • Prof. Galia Dafni at Concordia will teach a course in Fourier analysis next semester. Here is the draft course outline

  • 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).