Honors Analysis II - Spring 2013

MWF 10:40am - 11:30am, Jordan Hall of Science 322
Course website: www.nd.edu/~gszekely/HonorsAnalysisII.html

Instructor:

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

Office hours:

Wednesdays, Thursdays 4pm-5pm.

Textbook:

Kolmogorov, Fomin - Introductory Real Analysis

Grading:

Homework 30%; Quizzes 20%; Midterm 20%; Final 30%

Quizzes:

There will be a short quiz roughly every second Friday, starting Feb 1. 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.

Midterm:

There will be one midterm exam during class, on Wednesday February 27.

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 May 6, 6:15pm.

Homework:

There will be fortnightly homework assignments from the textbook, which can be found below along with the due date and time.

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. 16, 18 Measure in the plane Homework 1
due 1/25 in class
Jan. 21, 23, 25 Lebesgue measure; Measure on a semiring Homework 2
due 2/8 in class
Jan. 28, 30, Feb. 1 Countably additive measures; Extension of measures; Measurable functions Quiz 1 on Friday 2/1
Feb. 4, 6, 8 Measurable functions; Lebesgue integral Homework 3
due 2/22 in class
Feb. 11, 13, 15 Convergence theorems; Lebesgue vs. Riemann integral Quiz 2 on Friday 2/15
Feb. 18, 20, 22 Monotonic functions; Differentiation of monotonic functions Homework 4
due 3/8 in class
Feb. 25 Differentiation of an integral
Feb. 27 Midterm
Mar. 1 Functions of bounded variation
Mar. 4, 6, 8 Hilbert spaces Homework 5
due 3/27 in class
Mar. 11, 13, 15 Spring break
Mar. 18, 20, 22 Operators on Banach and Hilbert spaces Quiz 3 on Friday 3/22
Mar. 25, 27 Banach algebras Homework 6
due 4/12 in class
Mar. 29, Apr. 1 Easter break
Apr. 3, 5 C*-algebras Quiz 4 on Friday 4/5
Apr. 8, 10, 12 The Gelfand transform and Gelfand-Naimark theorem Homework 7
due 4/26 in class
Apr. 15, 17, 19 Spectral theorem for normal operators Quiz 5 on Friday 4/19
Apr. 22, 24, 26 Harmonic analysis on groups
Apr. 29, May 1 TBD
May 8 Final exam due 6:15pm