Office: 383M

Phone: 723-2226

E-mail: andras "at" math.stanford.edu

Course Assistant: Sander Kupers

Office: 381D

During the week of May 14-18, the office hours were held by Otis Chodosh (who is our WIM grader as well), in room 381D on T 11am-12:30pm, 2:15-3:45pm (same as regular), Th 9-11am and 1:15-3:15pm.

Class location: TTh 12:50-2:05pm, Room 380F.

Textbook: Johnsonbaugh and Pfaffenberger, Foundations of Mathematical Analysis.

The preliminary syllabus may change somewhat, but should give an indication of the scope and speed of the course.

This is an honors analysis course, emphasizing rigorous (i.e. logically careful) proofs and abstract notions, especially those related to metric spaces. It requires a thorough understanding of calculus, which provides the motivation for a significant part of the material. One of the goals of the course is to make you proficient in proof-writing, which is a crucial skill for more advanced mathematics courses, as well as a mechanism by which you can test your understanding of the material. Thus, there will be extensive homework assignments, and you should expect to spend much more time on this course than you are used to in other courses. In addition, there will be a writing assignment in the second half of the quarter which will have to be handed in in weekly installments. For a less abstract analysis class (focusing on Euclidean space), see Math 115.

Grading policy: The grade will be based on the weekly homework (20%), the writing assignment (15%), on the in-class midterm exam (25%) and on the in-class final exam (40%).

- A+: 180 and above,
- A: mid 150s-high 170s,
- A-: 140s-150ish,
- B+: mid 130s,
- B: mid 110s-low 130s,
- B-: 110ish,
- C+: 90s-low 100s,
- C: 70s-80s,
- C-: high 60s,
- D: low 60s and below.

Please try to arrive a few minutes early for the final so that we can start on time. Again, this is a closed book, closed notes, no calculators/computers, etc. exam.

The final covers every topic listed on the syllabus, plus the material on normed and inner product spaces we used to construct our examples, i.e. Sec 69-71 and Def. 72.1 from the book, as well as the material on the WIM assignment up to (including) Sec. 4. As far as integration is concerned, you only need to be familiar with it from the perspective of the WIM assignment.

The exam is of similar format to the midterm, except that it is longer, and you have more time. The length of the exam is a bit less than twice that of the midterm, and you have three hours, so hopefully the time pressure will be lower.

The homework will be due either in class or by 9pm in the instructor's mailbox on the designated day, usually Tuesdays. You are allowed to discuss the homework with others in the class, but you must write up your homework solution by yourself. Thus, you should understand the solution, and be able to reproduce it yourself. This ensures that, apart from satisfying a requirement for this class, you can solve the similar problems that are likely to arise on the exams.

The WIM assignment is posted, with the first installment due on Wednesday, May 16th. Please leave the assignments in the instructor's mailbox.

It is a closed book, closed notes, no calculators/computers, etc. exam.

Recommendations: please read through your class notes, the topics covered in the textbook (Chapters II-V, VI.30-31, 33, and VII.35-41), and make sure you know how to solve the homework problems, including the ones marked `do not hand in, but do'. In the exam, the instructions will state: "You may quote any theorem from the textbook or the lecture provided that you are not explicitly asked to prove it, and provided you state the theorem precisely and concisely (make sure to check the hypotheses when you quote a theorem)".

There will be at least one problem in which you will be asked to state a definition or a theorem in the first part (and then solve some problem related to it in the second half), so make sure you know all the definitions (such as least upper bound/greatest lower bound, countability, monotone sequence, limit, Cauchy sequence, lim sup/lim inf, sum of a series, absolutely/conditionally convergent, metric, open, closed sets, continuity, etc.) and major theorems (such as the positive integers are well ordered by <, theorems about countable sets, limit theorems, Bolzano-Weierstrass, Cauchy criterion for convergence, theorems about (absolute) convergence of series, power series, properties of open and closed sets, the relationship to continuity, etc.). Most exam problems will be similar to homework problems; and there will be at least one problem which is greatly similar to a `do not hand in' problem, so make sure you know how to solve those. In particular, do the problems from Sec. 40-41 (which are on the following week's problem set) before the exam to gain familiarity with these topics as well.

A practice exam (midterm given last year) is available with solutions. Recommendation: do all of your exam preparation before attempting the practice exam! You should expect five problems on the actual midterm as well.

- A+: mid-90s and above,
- A: 80s to low 90s,
- A-: high 70s,
- B+: low 70s,
- B: 60s,
- B-: 50s,
- C+: 40s,
- C: mid-high 30s,
- C-: low 30s,
- D: 20s and below.

- Problem Set 1, due Tuesday, April 10:
Hand in: 3.5, 3.8, 4.1, 5.4, 5.8, 6.6, 7.4, 7.9, 10.5, 10.9, 10.11, 11.5. Solutions, thanks to our CA Sander (and to Chris Henderson)!

Do not hand in, but do: 2.3, 2.4, 3.1, 4.4, 4.5, 6.3, 7.1, 7.5, 10.12, 11.3, 11.6, 11.11.

- Problem Set 2, due Wednesday, April 18:
Hand in: 9.7, 9.8, 9.10, 12.2, 12.6, 14.3, 15.2, 15.6, 16.3, 16.7, 16.8, 18.2, 18.3. Solutions, thanks to our CA Sander (and to Chris Henderson)!

Do not hand in, but do: 8.3, 9.6, 12.4, 13.1, 14.1, 14.2, 15.1, 15.4, 16.1, 16.10(a), 18.5.

- Problem Set 3, due Tuesday, April 24:
Hand in: 19.2, 20.7, 20.13, 20.21, 21.4, 22.3, 22.4, 23.1, 23.7, 24.3, 24.9. Solutions, thanks to our CA Sander (and to Chris Henderson)!

Do not hand in, but do: 19.1, 20.15, 20.16, 20.17, 21.2 (a),(b),(d),(e), 21.5, 22.1, 22.8, 23.3, 23.4, 23.5, 24.1, 24.4, 24.5, 24.6.

- Problem Set 4, due Tuesday, May 1:
Hand in: 16.11, 25.3, 26.1 (a)-(f), 26.4, 26.10, 27.1 (a)-(c), 27.2, 29.1, 29.3, 29.7, 29.14. Solutions, thanks to our CA Sander (and to Chris Henderson)!

Do not hand in, but do: 7.5, 7.6 (a), 26.5, 26.9, 27.3, 27.4, 29.2, 29.15.

- Problem Set 5, due Tuesday, May 8:
Hand in: 35.6, 35.7, 35.8, 35.9, 36.4, 37.7, 37.9(b), 37.10, 37.11, 38.8, 38.10, 38.13, 39.8, 39.10, 39.11, 39.12. Note that there is a typo in 38.13(e) at least in older editions of the text (as stated there, the result is trivial): what you need to show is that if Y is a closed subset of M and X is a subset of Y, then the set of limit points of X is a subset of Y (i.e. in the book the bar should be switched from the first of the two occurrences of X on this line to the second). This fact will be needed anyway to do 38.13(f). Solutions, thanks to our CA Sander (and to Chris Henderson)!

Do not hand in, but do: 35.2, 35.3, 35.5, 36.3, 36.5, 36.8, 36.9, 36.10, 37.2, 37.3, 37.4, 37.8, 38.2, 38.4, 38.9, 38.11, 39.4, 39.6, 39.9

- Problem Set 6, due Tuesday, May 15:
Hand in: 40.1, 40.7, 40.11, 40.17, 41.1, 41.5, 42.2, 42.3, 42.4, 42.6, 42.12. Solutions, thanks to our CA Sander (and to Chris Henderson)!

Do not hand in, but do: 40.2, 40.8, 40.9, 40.10, 40.13, 40.14, 40.15, 41.4, 42.1, 42.5, 42.7, 42.10.

- Problem Set 7, due Tuesday, May 22:
Hand in: 43.1, 43.3, 43.6, 43.7, 44.4, 44.7, 46.5, 46.7, 46.8, 46.11. Solutions, thanks to our CA Sander (and to Chris Henderson)!

Do not hand in, but do: 43.2, 44.2, 44.3, 46.6, 46.9.

- Problem Set 8, due Tuesday, May 29:
Hand in: 45.3, 45.6, 60.2, 60.4, 60.6, 69.5, 71.7. Solutions, thanks to our CA Sander (and to Chris Henderson)!

Do not hand in, but do: 45.1, 45.2, 60.1, 69.8, 71.10

- Problem Set 9, due Wednesday, June 6. Solutions, thanks to our CA Sander (and to Chris Henderson)!