Honors Analysis I - Fall 2012

MWF 10:40am - 11:30am, DeBartolo 119
Course website: www.nd.edu/~gszekely/HonorsAnalysis.html

Instructor:

Gábor Székelyhidi
Email: gszekely (at) nd.edu
Office: 277 Hurley
Tel: (574) 631-7412

Office hours:

Thursdays 3:15pm - 4:15pm.
Mondays 4pm - 5pm,
Feel free to email me if you would like to meet at some other time.

Textbook:

Kolmogorov, Fomin - Introductory Real Analysis.

This is a translation of the original book from Russian. A different translation of the same book has the title "Elements of the Theory of Functions and Functional Analysis". This alternative translation will be on reserve at the library, and you might find it helpful to consult it if you cannot follow something in "Introductory Real Analysis".

Grading:

Homework 20%; Quizzes 20%; Midterm 20%; Final 40%

Quizzes:

There will be a short quiz every second Friday, starting Aug 31. Each quiz will be on the proof of a result from the lectures. The proofs of the theorems in the course contain many useful techniques in analysis and so it is important to know them well. The goal of the quizzes is to help you keep up with learning the proofs throughout the semester. The lowest quiz grade will be dropped.

Midterms:

There will be one midterm exam during class, on Friday October 12.

Final:

I will give a take home final, on the last day of class. You will be able to spend 2 hours on it, and you can return it to me by Monday December 10, 6:15pm.

Homework:

There will be fortnightly written assignments which can be found below along with the due date and time. The solutions will be posted on Sakai.

Honesty:

This class follows the binding Code of Honor at Notre Dame. The graded work you do in this class must be your own. In the case where you collaborate with other students make sure to fairly attribute their contribution to your project.

Syllabus

List of theorems:

Theorems (11/10): This is a list of theorems you should know, and which you might need to prove on a quiz/exam. The list will grow as the course progresses. Here is the same list, with some sketched proofs for some of the theorems.

The dates given below for specific topics is only meant as a general guide, and the syllabus is likely to change as the course progresses.

Date Reading - page numbers in Kolmogorov-Fomin Homework/Quizzes
Aug. 22, 24 Sets; Functions
p. 1-6
Homework 1
due 9/7 in class
Aug. 27, 29, 31 Countable and uncountable sets; The real numbers
p. 9-15
Quiz 1 on Friday 8/31
Sep. 3, 5, 7 Metric spaces; Continuous maps; Limit points
p. 37-47
Homework 2
due 9/21 in class
Sep. 10, 12, 14 Convergence; Open and closed sets; Completeness
p. 47-60
Quiz 2 on Friday 9/14
Sep. 17, 19, 21 Baire's theorem; Completions; Contraction mappings
p. 61-73
Homework 3
due 10/5 in class
Sep. 24, 26, 28 Topological spaces; Continuity; Connectedness
p. 78-89
Quiz 3 on Friday 9/28
Oct. 1, 3, 5 Compactness; Arzela's theorem
p. 92-104
Homework 4
due 10/26 in class
Oct. 8, 10, 12 Real functions; Semicontinuity
p. 108-111
Midterm on Friday 10/12
Oct. 15, 17, 19 Fall break
Oct. 22, 24, 26 Linear spaces; Linear functionals; Convexity Homework 5
due 11/9 in class
Oct. 29, 31, Nov. 2 Zorn's Lemma; Hahn-Banach theorem
p. 20-28, p. 132-136
Quiz 4 on Friday 11/2
Nov. 5, 7, 9 Conjugate spaces, Hahn-Banach theorem Homework 6
due 11/30 in class
Nov. 12, 14, 16 Convexity, Weak convergence Quiz 5 on Friday 11/16
Nov. 19 Weak* convergence
Nov. 21, 23 Thanksgiving
Nov. 26, 28, 30 Axiom of choice, Zorn's lemma Quiz 6 on Friday 11/30
Dec. 3, 5 Applications
Dec. 10 Final exam: due 6:15pm