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.

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
Apr. 13, 15, 17 The Gelfand transform and Gelfand-Naimark theorem
Apr. 20, 22, 24 Spectral theorem for normal operators
Apr. 27, 29 Harmonic analysis on groups
Final exam