Honors Analysis II - Spring 2015
MWF 10:30am - 11:20am, Pasquerilla Center 109
Course website:
www.nd.edu/~gszekely/HonorsAnalysisII.html
Instructor:
Gábor Székelyhidi
Email:
gszekely (at) nd.edu
Office: 277 Hurley
Tel: (574) 631-7412
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 January 30. 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
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.
- 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(4/13): 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 in Kolmogorov-Fomin |
Homework/Quizzes |
| Jan. 14, 16 |
Measure in the plane |
Homework 1
due 1/26 in class |
| Jan. 19, 21, 23 |
Lebesgue measure; Measure on a semiring |
|
| Jan. 26, 28, 30 |
Countably additive measures; Extension of measures; Measurable functions |
Quiz 1 on Friday 1/30
Homework 2
due 2/9 in class |
| Feb. 2, 4, 6 |
Measurable functions; Lebesgue integral |
|
| Feb. 9, 11, 13 |
Convergence theorems; Lebesgue vs. Riemann integral |
Quiz 2 on Friday 2/13
Homework 3
due 2/23 in class |
| Feb. 16, 18, 20 |
Monotonic functions; Differentiation of monotonic functions |
|
| Feb. 23, 25, 27 |
Differentiation of an integral, Functions of bounded variation |
|
| Mar. 2 |
Hilbert spaces |
|
| Mar. 4 |
Midterm in class |
|
| Mar. 6 |
Hilbert spaces |
|
| Mar. 9, 11, 13 |
Spring break |
|
| Mar. 16, 18, 20 |
Operators on Banach and Hilbert spaces |
Homework 4
due 4/1 in class |
| Mar. 23, 25, 27 |
Banach algebras |
|
| Mar. 30, Apr. 1 |
C*-algebras |
|
| Apr. 2, Apr. 6 |
Easter break |
|
| Apr. 8, 10 |
C*-algebras |
Homework 5
due 4/24 in class |
| Apr. 13, 15, 17 |
The Gelfand transform and Gelfand-Naimark theorem |
Quiz 5 on Friday 4/17 |
| Apr. 20, 22, 24 |
Spectral theorem for normal operators |
|
| Apr. 27, 29 |
Harmonic analysis on groups |
|
| |
Final exam |
|