MATH 455: Honors Analysis 4

Winter 2020


Course web page: http://www.math.mcgill.ca/jakobson/courses/math455.html

Lectures:
  • Tuesday, Thursday 11:35-12:55, Burnside 1B39
  • The lectures will start on Tuesday, January 7.
  • The last lecture is on Tuesday, April 14, same time and room.
  • Due to university closures, there will be no lectures until the end of March. The course material will be uploaded to mycourses.
  • After March 30, the lectures will be online. Lecture notes will be posted on mycourses. Please, log on to mycourses for details.
  • Office hours: Tuesday/Thursday, 10:00-11:00, or by appointment.
  • Due to COVID-19, the office hours will be online instead of in-person. Further details will be announced on mycourses.
    • Instructor: D. Jakobson
    Office: BH1220
    Office Hours: Tuesday, Thursday, 10:00-11:00; or by appointment
    Tel: 398-3828
    E-mail: dmitry.jakobson AT mcgill.ca
    Web Page: www.math.mcgill.ca/jakobson
    Marker
  • Y. Cao
    • Description of BSc-MSc program
    Prerequisites: Math 454 or equivalent
    Texts:
  • 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.
    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 Thursday, January 23; pdf. Problems 13, 14, 15 may be turned in later, due date to be announced.
  • Assignment 2, due Thursday, February 6 pdf.
  • Hausdorff measure assignment, due date tba pdf.
  • A problem on Borel-Cantelli lemma, pdf.
  • Assignments will be updated on mycourses. Please, log on to mycourses
    • 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 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.
  • The take-home midterm will be distributed before the study break, and will be due on March 10 (the day of the first class after the break).
  • The class midterm will be held on March 12 (Thursday), 18:00-20:00, in Maass 10.
    • Final assignment:
  • Due to university closure, the final exam will be a take home final assignment. Please, submit your solutions electronically. The exam will be posted on April 20, and will be due on April 27.
  • 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) 20% Assignments + 30% Midterm + 50% Final assignment; OR (B) 20% Assignments + 80% Final assignment.
  • You may give a short presentation towards the end of the class (optional!) worth 10% of the mark. If you choose to give a presentation, your mark will be the best of (A); (B) or (C) 20% Assignments + 10% Presentation + 20% Midterm + 50% Final assignment.
  • Due to COVID-19, all in-person classes are cancelled for the Winter semester. Accordingly, the presentations will be given online. Please, prepare a short set of notes (handwritten and scanned, or typeset) for your presentation a few days in advance. Please, contact the instructor by email to reserve the time for your presentation.
    • MyCourses: Your scores on assignments, midterm, final, and your final mark will be posted on MyCourses
    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: helpdesk@math.mcgill.ca
    WEB LINKS in Calculus, Algebra, Geometry and Differential Equations.
    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, provided that there be timely communications to the students regarding the change.