**Course: MATH 131B, Analysis 2**, Lecture 1, Winter 2015

**Prerequisite:** MATH 131A, Analysis 1; MATH 115A, Linear algebra.

**Course Content:** This course is a continuation of MATH 131A. We
will treat the topics in real analysis from a more general perspective.
Topics include: metric spaces, point-set topology, function spaces,
convergence of sequences of functions, power series, analytic functions,
and Fourier analysis. This course should develop your ability
to write rigorous proofs.

*Last update:* 17 February
2015

**Instructor:** Steven Heilman, heilman(@-symbol)ucla.edu

**Office Hours:** Mondays, 10AM-12PM, Wednesdays, 11AM-12PM, MS
7370

**Lecture Meeting Time/Location:** Monday, Wednesday and Friday,
9AM-950AM, MS 5147

**TA:** Adam Azzam, adamazzam(@-symbol)gmail.com

**TA Office Hours:** Tuesdays 12PM-150PM, MS 6139

**TA Course Website:** here

**Discussion Session Meeting Time/Location:** Thursday, 9AM-950AM,
MS 5147

**Required Textbook:** Mathematical Analysis, 2nd Ed, T.
Apostol

**Other Textbooks (not required):** T. Tao, __Analysis II__, Hindustan Book Agency, 2009.

**First Midterm:** January 30th, 9AM-950AM, MS 5147

**Second Midterm:** February 23rd, 9AM-950AM, MS 5147

**Final Exam:** March 18, 8AM-11AM, MS 5147

**Other Resources:**
131BH,
Tao, Spring 2003: I would highly recommend
reading these lecture notes. My own lecture notes below are meant to be a more condensed presentation of similar material. So, if you
prefer a more thorough treatment, I recommend these notes (and the book). Note that we will probably not be
covering the Lebesgue intergral.

An
introduction to mathematical
arguments, Michael Hutchings,
An Introduction to Proofs,
How to Write Mathematical Arguments

**Exam Procedures:** Students must bring their UCLA ID cards to the
midterms and to the final exam. Phones must be turned off. Cheating on
an exam results in a score of zero on that exam. Exams can be
regraded at most 15 days after the date of the exam.

**Exam Resources:** Here
is a page with past exams for
the course. Here
is another page with past exams for the
course. Note that the content of these other courses may be slightly
different than ours.

**Homework Policy:**

- Late homework is not accepted.
- The lowest homework grade will be dropped. This policy is meant to account for illnesses, emergencies, etc.
- Do not submit homework via email.
- There will be 8 homework assignments, assigned weekly on Thursday and turned
in at the
**beginning**of each discussion session on the following Thursday - A random subset of the homework problems will be graded each week. However, it is strongly recommended that you try to complete the entire homework assignment.
- You may use whatever resources you want to do the homework, including computers, textbooks, friends, the TA, etc. However, I would discourage any over-reliance on search technology such as Google, since its overuse could degrade your learning experience. By the end of the quarter, you should be able to do the entire homework on your own, without any external help.
- All homework assignments must be
**written by you**, i.e. you cannot copy someone else's solution verbatim. I would encourage you to understand carefully how the homework solutions work, and how you would find such a solution on your own. Overusing collaborations or search technology should result in poor performance on the exams. - You are free to use any results in my lectures notes, up to the things that we have covered in class. If you use results from the book that are not in my notes, you must also produce the proof from the book, written in your own words.
- Label your homework with the lecture number and the discussion section number.
- Homework solutions will be posted ... (time)

- The final grade is given by the larger of the following two schemes. Scheme 1: homework (15%), the first midterm (20%), the second midterm (25%), and the final (40%). Scheme 2: homework (15%), largest midterm grade (35%), final (50%) The final grade will be curved. However, anyone who exceeds my expectations in the class by showing A-level performance on the exams and homeworks will receive an A for the class.
- We will use the MyUCLA gradebook.
- If you cannot attend one of the exams, you must notify me within the first two weeks of the start of the quarter. Later requests for rescheduling will most likely be denied.
- You must attend the final exam to pass the course.

** Tentative Schedule**: (This schedule may change slightly during the course.)

Week | Monday | Tuesday | Wednesday | Thursday | Friday |

1 | Jan 5: 3.13, Metric Spaces | Jan 7: 4.2, Convergence of sequences | Jan 8: No homework due | Jan 9: 3.14, Topology of metric spaces | |

2 | Jan 12: 3.15, Compact sets | Jan 14: 4.8, 4.12, Continuous functions on metric spaces | Jan 15: Homework 1 due | Jan 16: 4.13, 4.14, Continuity and compactness | |

3 | Jan 19: No class | Jan 21: 4.16, 4.17, 4.18 Connectedness | Jan 22: Homework 2 due | Jan 23: 9.1, 9.2, Sequences and Series of Functions | |

4 | Jan 26: 9.3, 9.4, Uniform Convergence and continuity | Jan 28: 9.8, Uniform convergence and integration | Jan 29: Homework 3 due | Jan 30: Midterm #1 | |

5 | Feb 2: 9.10, Uniform convergence and differentiation | Feb 4: 11.15, 9.14, Approximation by polynomials, power series | Feb 5: Homework 4 due | Feb 6: 9.14, 9.15, 9.18, 9.19, More power series | |

6 | Feb 9: Exponential and Logarithm | Feb 11: 1.21-1.26, 1.32, Trigonometric Functions | Feb 12: Homework 5 due | Feb 13: 11.1, Periodic Functions | |

7 | Feb 16: No class | Feb 18: 11.2, Inner products on periodic functions | Feb 19: Homework 6 due | Feb 20: Trigonometric Polynomials | |

8 | Feb 23: Midterm #2 | Feb 25: 11.3, 11.15, Approximation by trigonometric polynomials | Feb 26: No homework due | Feb 27: 11.3, Fourier inversion and Plancherel theorems | |

9 | Mar 2: 12.1, Differentiation in several variables | Mar 4: 12.2, 12.9, Directional Derivatives, Chain Rule | Mar 5: Homework 7 due | Mar 6: 12.13, Clairaut's Theorem | |

10 | Mar 9: Leeway/review | Mar 11: Leeway/review | Mar 12: Homework 8 due | Mar 13: Review of course |

**Advice on succeeding in a math class:**

- Review the relevant course material
**before**you come to lecture. Consider reviewing course material a week or two before the semester starts. - When reading mathematics, use a pencil and paper to sketch the calculations that are performed by the author.
- Come to class with questions, so you can get more out of the
lecture. Also, finish your homework at
least
**two days**before it is due, to alleviate deadline stress. - Write a rough draft and a separate final draft for your homework. This procedure will help you catch mistakes. Also, consider typesetting your homework. Here is a template .tex file if you want to get started typesetting.
- If you are having difficulty with the material or a particular homework problem, review Polya's Problem Solving Strategies, and come to office hours.