Mathematics 131AH - Honors Analysis - Fall 2016 - UCLA

Time: 11-11:50 MWF, plus 11-11:50 Tu for discussion.
Place: Lectures: MS 5138. Discussion: MS 6627.

Instructor: Burt Totaro.

Course web page:

Office hours: 3-3:50 M and 2-2:50 F, in my office, MS 6136.
TA: Dimitrios Ntalampekos (
TA office hour: 1:00-2:00 F in MS 3975.

Book: W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, third ed., ISBN: 978-0070542358.

Material to be covered: Rigorous treatment of the foundations of real analysis, including construction of the rationals and reals; metric space topology, including compactness and its consequences; numerical sequences and series; continuity, including connections with compactness; rigorous treatment of the main theorems of differential calculus. That amounts to Chapters 1-4 of Rudin, plus some of Chapter 5 as time permits.

Homework will be due each week in discussion, and returned the next week. There will be no makeup or late homework accepted, but the lowest homework grade will be dropped.

In this class you will be required to write precise mathematical statements in a clear logical order, and present pictures or examples as necessary to illustrate your work. Acquiring these skills is impossible without steady practice: it is essential that you do the homework problems carefully and promptly.

You may discuss homework problems with other students, the TA, or me, before they are turned in. I do expect, though, that: (i) you should make a serious effort to do the exercise yourself before discussing it with anyone, and (ii) you should write up the solution yourself after understanding it thoroughly, without following someone else's written version. Otherwise, homework will not help you to prepare for the exams. Identical solutions to a source will get zero credit.

Homework 1. Due Tuesday, September 27.
Homework 2. Due Tuesday, October 4.
Homework 3. Due Tuesday, October 11.
Homework 4. Due Tuesday, October 25.
Homework 5. Due Tuesday, November 1.
Homework 6. Due Tuesday, November 8.
Homework 7. Due Tuesday, November 22.
Homework 8. Due Tuesday, November 29.

Midterm Exams: We will have two midterm exams. The dates are Monday, October 17 and Monday, November 14. There will be no makeup exams.
Sample Midterm 2.

Final Exam: The final exam is on Friday, December 9, 2016 from 11:30 AM to 2:30 PM. You must take the final to pass the class! If you have a documented reason that you are unable to take the final, you will receive an Incomplete.

Evaluation: Grading: Grades will be assigned based on the higher of the following two schemes:
10% homework + 25% first midterm + 25% second midterm + 40% final
10% homework + 35% (best of two midterms) + 55% final Miscellanea: Catalog description: 131A. Analysis (Honors). (4) Lecture, three hours; discussion, one hour. Required: courses 32B and 33B, with grades of B or better. Recommended: course 115A. Honors sequence parallel to course 131A. P/NP or letter grading. Rigorous introduction to foundations of real analysis; real numbers, point set topology in Euclidean space, functions, continuity.

Tentative schedule of lectures, in terms of the book:
9/23: Ch. 1. Mathematical logic, induction.
9/26: Ordered fields. 9/28: Axioms for R; supremum. 9/30: Applications of the axioms.
10/3: Complex numbers, Euclidean spaces. 10/5: Construction of R. 10/7: Ch. 2. Finite and infinite sets.
10/10: Countable sets. 10/12: Metric spaces, open and closed sets. 10/14: Compact sets.
10/17: Midterm 1. 10/19: More on compactness. 10/21: Perfect sets. Connected sets.
10/24: More on connectedness. 10/26: Ch. 3. Convergent sequences. 10/28: Subsequences, Cauchy sequences, completeness.
10/31: Lim sup, lim inf. 11/2: Some special sequences. 11/4: Series.
11/7: Absolute convergence. 11/9: Ch. 4. Limits of functions. 11/11: Veterans Day holiday.
11/14: Midterm 2. 11/16: Continuous functions. 11/18: Continuity and compactness.
11/21: Continuity and connectedness. 11/23: Ch. 5. The derivative. 11/25: Thanksgiving holiday.
11/28: Chain rule. Mean value theorem. 11/30: Taylor's theorem. 12/2: Review.