MATH 455: Honors Analysis 4

Winter 2021

Course web page:

  • Tuesday, Thursday, 14:30-16:00, for a moment on zoom (but in class lectures possible later on), further details to be announced.
  • The lectures will start on Thursday, January 7.
  • The lectures are scheduled to end on Tuesday, April 13.
  • Office hours (tentative): Wednesday, 15:00-16:00; Thursday, 16:00-17:00; or by appointment.

  • Instructor: D. Jakobson
    Office: BH1220
    Tel: 398-3828
    E-mail: dmitry.jakobson AT
    Web Page:
  • Karim Elmallakh

  • Description of BSc-MSc program

    Prerequisites: Math 454 or equivalent
  • Real Analysis. Measure, Integration and Hilbert spaces, by E. Stein and R. Shakarchi.
  • Supplementary: Real Analysis, 4th edition, by H.L. Royden and P.M. Fitzpatrick.
  • Supplementary: 2010 Lecture notes taken by Robert Gibson.
  • Supplementary: Notes on Introductory Point-Set Topology by Allen Hatcher.

  • Syllabus: L^p space, Duality, weak convergence; inequalities of Young, Holder and Minkowski. Point-set topology: topological space, dense sets, completeness, compactness, connectedness and path-connectedness, separability. Tychnoff theorem, Stone-Weierstrass theorem. Arzela-Ascoli, Baire category theorem, Open mapping theorem, Closed Graph theorem, Uniform Boundedness principle.
    Math 455, Winter 2020
    Assignments: There will be be several assignments. Due dates will be announced on mycourses and on the course web page. There will be penalty for late assignments, except in cases of emergency. Depending on availability of markers and other factors, some problems may not be marked.
    Handouts (for previous years/different classes!):
  • 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 crowdmark (timed) or a 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.
  • The take-home midterm will be distributed before the study break, and will be due shortly after the end of the break.
  • The time window for the crowdmark midterm will be announced on Mycourses.

  • Final assignment:
  • The final exam will be a take home final assignment, details to be announced on mycourses.
  • 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.

  • Grading
  • Your final mark will be the largest of the following: (A) 30% Assignments + 30% Midterm + 40% Final assignment; OR (B) 30% Assignments + 70% Final assignment.
  • You may give an optional short presentation towards the end of the class worth 10% of the mark. If you choose to give a presentation, your mark will be the best of (A); (B) or (C) 30% Assignments + 10% Presentation + 30% Midterm + 30% Final assignment.

  • MyCourses: Your scores on assignments, midterm, final, and your final mark will be posted on MyCourses
    Course material from previous courses at McGill:
  • D. Jakobson. Math 455, 2020
  • 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.
    HELPDESK and their email:
    Health and Wellness Resources at McGill: Student well-being is a priority for the University. All of our health and wellness resources have been integrated into a single Student Wellness Hub, your one-stop shop for everything related to your physical and mental health. If you need to access services or get more information, visit the Virtual Hub at or drop by the Brown Student Services Building (downtown) or Centennial Centre (Macdonald campus). Within your faculty, you can also connect with your Local Wellness Advisor (to make an appointment, visit
    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, provided that there be timely communications to the students regarding the change.