MATH 354: Honors Analysis 3

Fall 2006


Course web page: http://www.math.mcgill.ca/jakobson/courses/math354.html

Lectures: MWF 9:30-10:30, Burnside 920
Tuesday, October 10, 9:30-10:30, Burnside 920
Review lecture: Wednesday, November 29, 17:00-18:00, Burnside 920
Instructor: D. Jakobson
Office: BH1212, Office Hours: Mon 11:00-12:00, Wed. 10:30-11:30, or by appointment
Tel: 398-3828
E-mail: jakobson AT math.mcgill.ca
Web Page: www.math.mcgill.ca/jakobson
Markers: Darren Swersky and Sharam Shahlaei-Far
Tutorials (starting in October): Gabriel Gauthier
Time (alternate weeks): Friday, 11:30-12:30, Burnside 1214
OR: Thursday, 1pm-2pm, Burnside 1234
Week of November 13 only: also Tuesday, 1pm-2pm, Burnside 1234
Monday, December 4 only: 10:30-12:30, Burnside 1234
Gabriel's times at Helpdesk: Monday, 2pm - 5pm; Thursday, 2:30pm - 5pm; Friday - 1pm - 2:30pm.
Wednesday, December 13 (Jakobson), 11:00am, Burnside 920
Thursday, December 14 (Gauthier), 11:00am, Burnside 1205
Prerequisites: Math 255 or equivalent
Text: Introductory real analysis, by A. Kolmogorov and S. Fomin, Dover Publications (on reserve in Schulich library).
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 Wednesday, September 27 (extended!) ps and pdf. Solutions: ps and pdf; continued: ps and pdf
  • Assignment 2, due Friday, October 13 (extended!) ps and pdf. Solutions: ps and pdf
  • Assignment 3, due Friday, November 10 (extended). ps and pdf. Problem 1 (iii) is extra credit. In Problem 6, prove that the set has COMPACT CLOSURE (in the space of continuous functions with uniform distance). Solutions: ps and pdf
  • Assignment 4, due Monday, November 20 (extended). ps and pdf. Problems 1 and 3 are optional. Solutions: ps and pdf
  • Assignment 5, due Wednesday, November 29 (Corrected!) ps and pdf. Problems 4 and 8 are optional, Problem 9 is extra credit. Problems 6 and 7 are due on Monday, December 4. In Problem 2, only find the first derivative. In Problem 4, you can assume that x is greater than or equal to 0. Solutions I: ps and pdf. Solutions II: ps and pdf
  • Practice problems (not for credit): ps and pdf. Solutions: ps and pdf
  • Practice problems II (not for credit), with solutions: ps and pdf
  • 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: 1. ps and pdf

    • Midterm: There will be an in-class midterm, October 23. It will be held at the same time and place as the regular lecture. Solutions: ps and pdf
    Old Midterm solutions (pdf)
    Missed midterm cannot be redone. If you miss the midterm for any reason, the weights for your mark will be: Assignments 20%, Final 80%.
    Final: There will be a three hour final exam. Date: December 18, 2:00 pm

    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:
  • 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 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
  • Harmonic Analysis page by Terry Tao (there is a lot of advanced stuff there)
    • HELPDESK and their email: helpdesk@math.mcgill.ca
    WEB LINKS in Calculus, Algebra, Geometry and Differential Equations.
    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).