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) 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 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.

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
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
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
Sep. 29, Oct. 1, 3 Topological spaces; Continuity; Connectedness Quiz 3 on Friday 10/3
Oct. 6, 8, 10 Compactness; Arzela's theorem Homework 4
Oct. 13, 15, 17 Real functions; Semicontinuity Midterm on Friday 10/17
Oct. 20, 22, 24 Fall break
Oct. 27, 29, 31 Linear spaces; Linear functionals; Convexity Homework 5
Nov. 3, 5, 7 Zorn's Lemma; Hahn-Banach theorem Quiz 4 on Friday 11/7
Nov. 10, 12, 14 Conjugate spaces, Hahn-Banach theorem Homework 6
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