Lectures MWF 9:10-10:00, 247 Cory Hall

Office hours F 16:00-18:00, 821 Evans Hall, and by appt.

- 08/30: The language of set theory.
- 09/02: Ordered sets.
- 09/04: Cardinality.
- 09/06: Metrics.
- 09/09: Metric spaces.
- 09/11: Compactness.
- 09/13: Sigma-algebras.
- 09/16: Measures.
- 09/18: Outer measures.
- 09/20: Measures, premeasures and outer measures: connections.
- 09/23: Borel measures on R.
- 09/25: Lebesgue measure. Cantor sets.
- 09/27: Measurable functions.
- 09/30: Simple functions as approximants of measurable functions.
- 10/02: Integration of simple functions.
- 10/04: The monotone convergence theorem.
- 10/07: Integration of nonnegative functions.
- 10/09: Integration of real-valued functions.
- 10/11: Integration of complex-valued functions.
- 10/14: Midterm.
- 10/16: Discussion of the midterm.
- 10/18: The dominated convergence theorem and consequences.
- 10/21: Modes of convergence.
- 10/23: Egoroff's Theorem.
- 10/25: Signed measures. Hahn decomposition.
- 10/28: Jordan decomposition. Absolute continuity.
- 10/30: The Lebesgue-Radon-Nikodym Theorem I.
- 11/01: Lebesgue-Radon-Nikodym II.
- 11/04: Complex measures.
- 11/06: Differentiation on R.
- 11/08: Functions of bounded variation.
- 11/13: Topological spaces.
- 11/15: Countability and separation axioms.
- 11/18: Continuous maps.
- 11/20: Urysohn's Lemma and Tietze Extension Theorem.
- 11/22: Nets.
- 11/25: Compact spaces.
- 11/27: Locally compact Hausdorff spaces.
- 12/02: Functions on LCH spaces.
- 12/04: The Arzela-Ascoli theorem.
- 12/06: The Stone-Weierstrass Theorem.
- 12/09: Introduction to L^p spaces.

- Books.
- Gerald B. Folland, Real analysis. Modern techniques and their applications, Second edition. Wiley, 1999.
- Walter Rudin, Real and complex analysis, Third edition, McGrew-Hill, 1987.
- Walter Rudin, Functional analysis, Second edition, McGraw-Hill, 1991.

The first book will be the main textbook for the course.

The instructor welcomes cooperation among students and the use of books.
However, handing in homework that makes use of other people's work (be it from a fellow
student, a book or paper, or whatever) **without** explicit
acknowledgement is considered academic misconduct.

- Homework assignment #1, due Sep 9th.
- Homework assignment #2, due Sep 16th.
- Homework assignment #3, due Sep 23rd: Exercises 10, 13, 14, 17, 19, 23 from Chapter 1 of Folland.
- Homework assignment #4, due Sep 30th: Exercises 27, 28, 29, 32 from Chapter 1, Exercises 2, 3 from Chapter 2 of Folland.
- Homework assignment #5, die Oct 7th: Exercises 8, 13, 15, 17 from Chapter 2 of Folland.
- Homework assignment #6, due Oct 14th: Exercises 12, 18 from Chapter 2 of Folland.
- Homework assignment #7, due Oct 21st: Exercises 20, 22, 27 from Chapter 2 of Folland.
- Homework assignment #8, due Oct 28th.
- Homework assignment #9, due Nov 4th.
- Homework assignment #10, due Nov 13th: Exercises 18, 20, 27, 30, 37, 38 from Chapter 3 of Folland.
- Homework assignment #11, due Nov 18th: Exercises 4, 7, 8, 13 from Chapter 4 of Folland.
- Homework assignment #12, due Nov 25th: Exercises 16, 21, 23, 29 from Chapter 4 of Folland.
- Homework assignment #13, due Dec 2nd: Exercises 30, 32, 33, 39, 42 from Chapter 4 of Folland.
- Homework assignment #14, due Dec 9th: Exercises 46, 54, 63, 64, 70a-c from Chapter 4 of Folland.

Solutions to selected homework problems have been posted on Piazza.

There is one midterm Monday October 14th in class.

Here is a final test.

Last modified: Dec 13, 2013