18.100B Spring Semester, 2002

General Information

Class meetings: Monday, Wednesday, and Friday 2:00–3:00, in 4-163.

Text: Walter Rudin, Principles of Mathematical Analysis, third edition. You should try to read the text before class as well as after. Both your own understanding and your chance of catching the lecturer in a faux pas will be greatly increased.

Lecturer: David Vogan, 2-284. Telephone: 253-4991. E-mail: dav@math.mit.edu. My office hours are Wednesday 3–4, Thursday 3–4, or by appointment.

Homework will be assigned in most classes. Problems assigned during each week will be collected at the beginning of the first class of the following week. One of the goals of the course is for you to learn to write clear, complete, and elegant proofs (in that order, perhaps). The problem sets are the best place to show your accomplishments without time pressure.

Exams: There will be three one-hour exams during the lecture hour, on March 4, March 22, and April 19. There will be a three-hour final exam during finals week. The exams will all be closed book.

Grading: Each hour exam will be worth 100 points, the final exam will be worth 200 points, and the graded problem sets will be worth 20 points each.

Schedule

Wed 2/6 Lecture 1 1–8  Ordered fields
Fri 2/8  Lecture 2 8–15  Real and complex fields
Mon 2/11 Lecture 3 16–17  Euclidean spaces
Wed 2/13 Lecture 4 24–30  Functions and infinite sets
Fri 2/15 Lecture 5 30–34  Metric spaces
Tues 2/19 Lecture 6 34–36  Metric spaces
Wed 2/20 Lecture 7 36–38  Compact sets
Fri 2/22 Lecture 8 38–40  Heine-Borel theorem
Mon 2/25 Lecture 9 40–42  Sequences and compact sets
Wed 2/27 Lecture 10 42–43  Connected sets
Fri 3/1  Lecture 11 1–43  Review
Mon 3/4 Lecture 12  Exam 1 on Chapters 1–2
Wed 3/6 Lecture 13 47–52  Convergent sequences
Fri 3/8  Lecture 14 52–57  Cauchy sequences
Mon 3/11 Lecture 15 57–65  Series, comparison test
Wed 3/13 Lecture 16 65–72  Root and ratio tests
Fri 3/15 Lecture 17 83–89  Continuous functions
Mon 3/18 Lecture 18 89–93  Continuity and compactness
Wed 3/20 Lecture 19 93–97  Continuity and connectedness
Fri 3/22 Lecture 20  Exam 2 on Chapters 3–4
3/25–3/29 Spring Vacation

Mon 4/1 Lecture 21 103–106 Differentiable functions
Wed 4/3 Lecture 22 107–110 Mean value theorems, L’Hôpital’s rule
Fri 4/5  Lecture 23 110–113 Taylor’s theorem
Mon 4/8 Lecture 24 120–125 Riemann integrals
Wed 4/10 Lecture 25 125–129 Integrability criteria
Fri 4/12 Lecture 26 132–136 Integration and differentiation
Mon 4/15 Holiday
Wed 4/17 Lecture 27 1–136  Review
Fri 4/19 Lecture 28  Exam on chapters 5–6
Mon 4/22 Lecture 29 143–147 Sequences and series of functions
Wed 4/24 Lecture 30 147–148 Uniform convergence
Fri 4/26 Lecture 31 149–151 Uniform convergence and continuity
Mon 4/29 Lecture 32 151–154 Uniform convergence and calculus
Wed 5/1 Lecture 33 159–161 Stone-Weierstrass theorem
Fri 5/3  Lecture 34 161–165 Stone-Weierstrass theorem
Mon 5/6 Lecture 35 172–178 Power series
Wed 5/8 Lecture 36 178–184 Exponential and trig functions
Fri 5/10 Lecture 37 185–192 Fourier series
Mon 5/13 Lecture 38  Uniform distribution of fractional parts
Wed 5/15 Lecture 39 1–192  Review
week of 5/20–5/24  Final Exam on chapters 1–8