Real Analysis 18.100B

Fall 2011

Lecturer: Kiril Datchev, room 2-230, datchev@math.mit.edu.

Class meetings: Tuesdays and Thursdays 1:00-2:30 in room 4-163.

Textbook: Walter Rudin, Principles of Mathematical Analysis.

Recommended reading: G. H. Hardy, A Course of Pure Mathematics. Edmund Landau, Foundations of Analysis. Tom M. Apostol, Mathematical Analysis.

Office hours: My office hours are Tuesdays 2:30-4:00 in room 2-101 and Wednesdays 10:00-11:00 in room 2-102. The teaching assistant, Michael Andrews, has office hours Wednesdays 5:30-7:30 in room 2-492. Students are also welcome to attend the office hours of Hamid Hezari and Kyle Ormsby, which are listed here.

Grading is based on:

ten problem sets, worth 21 points each, due on 9/15, 9/22, 9/29, 10/13, 10/20, 10/27, 11/3, 11/17, 12/1, 12/8,
two midterms, worth 100 points each, one on October 4th and one on November 8th,
one final exam, worth 200 points, 1:30-4:30 on December 20th.

Problem sets are due Thursdays at 4:00 in room 2-108. Late problem sets are not accepted. Hand in the three parts of the assignment separately, and indicate whether you are registered for 18.100B or 18.100C on each part.

Problem set 1 due September 15th. Solutions.
Problem set 2 due September 22nd. Solutions.
Problem set 3 due September 29th. Solutions.
Review sheet for the first midterm.
Problem set 4 due October 13th. Solutions.
Problem set 5 due October 20th. Solutions.
Problem set 6 due October 27th. Solutions.
Problem set 7 due November 3rd. Solutions.
Review sheet for the second midterm.
Problem set 8 due November 17th. Solutions.
Problem set 9 due December 1st. Solutions.
Problem set 10 due December 8th. Solutions.
Review sheet for the final.

There will be many extra office hours for the week from Tuesday 12/13 to Monday 12/19: I will have office hours Tuesday 2:30-4 in 2-101, Wednesday 10-11 in 2-102, and Monday 4:30-6:30 in 2-102. Michael will have office hours Wednesday, Thursday, Friday, Saturday and Sunday 11:30-1 in 2-492. Hamid will have office hours Tuesday 11-12 in 2-363B, Wednesday 1-2 in 2-363B, and Monday 6:30-8:30 in 2-102. Phew!

Schedule

Date Pages Topics

9/8 1 – 12 ordered sets, fields, real numbers

9/13 12 – 26 complex numbers, Euclidean spaces, functions, finite and infinite sets

9/15 26 – 34 countable and uncountable sets, metric spaces

9/20 34 – 36 open and closed sets

9/22 36 – 38 compact sets

9/27 38 – 40 the Heine-Borel theorem, the Bolzano-Weierstrass theorem

9/29 40 – 43 perfect sets, the Cantor set, connected sets

10/4 1 – 46 Midterm on Chapters 1 and 2

10/6 47 – 54 sequences, convergence, Cauchy sequences

10/13 54 – 60 completeness, monotonic sequences, upper and lower limits, series, comparison test

10/18 60 – 65 series of nonnegative terms, the number e

10/20 65 – 72 root and ratio tests, power series, conditional and absolute convergence

10/25 83 – 90 continuous functions, continuity and compactness

10/27 90 – 97 uniform continuity, continuity and connectedness, monotonic functions

11/1 103 – 109 differentiation, mean value theorems

11/3 109 – 113 l'Hospital's rule, Taylor's theorem

11/8 47 – 119 Midterm on Chapters 3, 4 and 5

11/10 120 – 127 the Riemann-Stieltjes integral

11/15 128 – 137 properties of the integral, fundamental theorem of calculus, rectifiable curves

11/17 140 – 148 sequences and series of functions, pointwise and uniform convergence

11/22 148 – 154 uniform convergence, continuity and differentiation

11/29 154 – 158 equicontinuous families of functions, the Arzelà-Ascoli theorem

12/1 159 – 165 the Stone-Weierstrass theorem

12/6 172 – 178 functions given by power series

12/8 178 – 184 exponential, logarithmic and trigonometric functions

12/13 185 – 192 Fourier series

12/20 1 – 192 Final Exam on Chapters 1 – 8

Schedule
Date	Pages	Topics
9/8	1 – 12	ordered sets, fields, real numbers
9/13	12 – 26	complex numbers, Euclidean spaces, functions, finite and infinite sets
9/15	26 – 34	countable and uncountable sets, metric spaces
9/20	34 – 36	open and closed sets
9/22	36 – 38	compact sets
9/27	38 – 40	the Heine-Borel theorem, the Bolzano-Weierstrass theorem
9/29	40 – 43	perfect sets, the Cantor set, connected sets
10/4	1 – 46	Midterm on Chapters 1 and 2
10/6	47 – 54	sequences, convergence, Cauchy sequences
10/13	54 – 60	completeness, monotonic sequences, upper and lower limits, series, comparison test
10/18	60 – 65	series of nonnegative terms, the number e
10/20	65 – 72	root and ratio tests, power series, conditional and absolute convergence
10/25	83 – 90	continuous functions, continuity and compactness
10/27	90 – 97	uniform continuity, continuity and connectedness, monotonic functions
11/1	103 – 109	differentiation, mean value theorems
11/3	109 – 113	l'Hospital's rule, Taylor's theorem
11/8	47 – 119	Midterm on Chapters 3, 4 and 5
11/10	120 – 127	the Riemann-Stieltjes integral
11/15	128 – 137	properties of the integral, fundamental theorem of calculus, rectifiable curves
11/17	140 – 148	sequences and series of functions, pointwise and uniform convergence
11/22	148 – 154	uniform convergence, continuity and differentiation
11/29	154 – 158	equicontinuous families of functions, the Arzelà-Ascoli theorem
12/1	159 – 165	the Stone-Weierstrass theorem
12/6	172 – 178	functions given by power series
12/8	178 – 184	exponential, logarithmic and trigonometric functions
12/13	185 – 192	Fourier series
12/20	1 – 192	Final Exam on Chapters 1 – 8