Honors Analysis I  Fall 2014
MWF 10:30am  11:20am, Pasquerilla Center 102
Course website:
www.nd.edu/~gszekely/HonorsAnalysis.html
Instructor:
Gábor Székelyhidi
Email:
gszekely (at) nd.edu
Office: 277 Hurley
Tel: (574) 6317412
Textbook:
Kolmogorov, Fomin  Elements of the Theory of Functions and
Functional Analysis.
Note that this is different from the book called
"Introductory Real Analysis", by the same authors, although both are
translations of the same Russian original.
Grading:
Homework 30%; Quizzes 20%; Midterm 20%; Final 30%
Quizzes:
There will be a short quiz
every second Friday,
starting September 5. 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 17.
Final:
I will give a take home final, on the last day of class.
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. 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 
Homework/Quizzes 
Aug. 27, 29 
Sets; Functions 
Homework 1 due 9/8 in class

Sep. 1, 3, 5 
Countable and uncountable sets; The real numbers 
Quiz 1 on Friday 9/5 
Sep. 8, 10, 12 
Metric spaces; Continuous maps; Limit points 
Homework 2 due 9/22 in class

Sep. 15, 17, 19 
Convergence; Open and closed sets; Completeness 
Quiz 2 on Friday 9/19 
Sep. 22, 24, 26 
Baire's theorem; Completions; Contraction mappings 
Homework 3 due 10/6 in class

Sep. 29, Oct. 1, 3 
Topological spaces; Continuity; Connectedness

Quiz 3 on Friday 10/3 
Oct. 6, 8, 10 
Compactness; Arzela's theorem 

Oct. 13, 15, 17 
Real functions; Semicontinuity 
Midterm on Friday 10/17 
Oct. 20, 22, 24 
Fall break 
Homework 4 due 11/3 in class 
Oct. 27, 29, 31 
Linear spaces; Linear functionals; Convexity 
Homework 5 due 11/17 in class

Nov. 3, 5, 7 
Zorn's Lemma; HahnBanach theorem 
Quiz 4 on Friday 11/7 
Nov. 10, 12, 14 
Conjugate spaces, HahnBanach theorem 
Homework 6 due 12/8 in class

Nov. 17, 19, 21 
Convexity, Weak convergence 
Quiz 5 on Friday 11/21 
Nov. 24 
Weak* convergence 

Nov. 26, 28 
Thanksgiving 

Dec. 1, 3, 5 
Axiom of choice, Zorn's lemma 
Quiz 6 on Friday 12/5 
Dec. 8, 10 
Applications 


Final exam 
