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) 6317412
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.
 Late homework will not be accepted.
 The lowest homework grade will be dropped.
 Please staple or paper clip your work.
 Don't forget to write your name on it!
 You may ask others for help with your homework. However, it is
unwise to do the homework exclusively in a group; there is no substitute for
the insight and self confidence that comes from successful individual study.
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 KolmogorovFomin 
Homework/Quizzes 
Aug. 22, 24 
Sets; Functions p. 16 
Homework 1
due 9/7 in class 
Aug. 27, 29, 31 
Countable and uncountable sets; The real numbers p. 915 
Quiz 1 on Friday 8/31 
Sep. 3, 5, 7 
Metric spaces; Continuous maps; Limit points p. 3747 
Homework 2
due 9/21 in class 
Sep. 10, 12, 14 
Convergence; Open and closed sets; Completeness p. 4760 
Quiz 2 on Friday 9/14 
Sep. 17, 19, 21 
Baire's theorem; Completions; Contraction mappings p. 6173 
Homework 3
due 10/5 in class 
Sep. 24, 26, 28 
Topological spaces; Continuity; Connectedness
p. 7889 
Quiz 3 on Friday 9/28 
Oct. 1, 3, 5 
Compactness; Arzela's theorem p. 92104 
Homework 4
due 10/26 in class 
Oct. 8, 10, 12 
Real functions; Semicontinuity p. 108111 
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; HahnBanach theorem p. 2028, p. 132136 
Quiz 4 on Friday 11/2 
Nov. 5, 7, 9 
Conjugate spaces, HahnBanach 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 
