Honors Analysis I - Fall 2012
MWF 10:40am - 11:30am, DeBartolo 119
Course website: www.nd.edu/~gszekely/HonorsAnalysis.html
Email: gszekely (at) nd.edu
Office: 277 Hurley
Tel: (574) 631-7412
Thursdays 3:15pm - 4:15pm.
Mondays 4pm - 5pm,
Feel free to email me if you would like to meet at some other time.
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".
Homework 20%; Quizzes 20%; Midterm 20%; Final 40%
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
There will be one midterm exam during class, on
Friday October 12.
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
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.
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.
List of 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.
|| Reading - page numbers in Kolmogorov-Fomin
| Aug. 22, 24
|| Sets; Functions
| Homework 1
due 9/7 in class
| Aug. 27, 29, 31
|| Countable and uncountable sets; The real numbers
| Quiz 1 on Friday 8/31
| Sep. 3, 5, 7
|| Metric spaces; Continuous maps; Limit points
| Homework 2
due 9/21 in class
| Sep. 10, 12, 14
|| Convergence; Open and closed sets; Completeness
| Quiz 2 on Friday 9/14
| Sep. 17, 19, 21
|| Baire's theorem; Completions; Contraction mappings
| Homework 3
due 10/5 in class
| Sep. 24, 26, 28
|| Topological spaces; Continuity; Connectedness
| Quiz 3 on Friday 9/28
| Oct. 1, 3, 5
|| Compactness; Arzela's theorem
| Homework 4
due 10/26 in class
| Oct. 8, 10, 12
|| Real functions; Semicontinuity
| 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
| Nov. 26, 28, 30
|| Axiom of choice, Zorn's lemma
|| Quiz 6 on Friday 11/30
| Dec. 3, 5
| Dec. 10
|| Final exam: due 6:15pm