Real Analysis 18.100B

Fall 2012

Lecturer: Kiril Datchev, room 2-173, datchev@math.mit.edu.

Class meetings: Tuesdays and Thursdays 9:30-11:00 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. See also these notes on the text by George Bergman. For inspirational reading, consult The Study of Mathematics by Bertrand Russell.

Grading is based on:

ten problem sets, worth 20 points each, due on 9/13, 9/20, 9/27, 10/11, 10/18, 10/25, 11/1, 11/15, 11/29, 12/6,
two midterms, worth 100 points each, one on October 2nd and one on November 6th,
one final exam, worth 200 points, on Thursday, December 20th, 1:30pm-4:30pm in Johnson Track (upstairs).

Problem sets are due Thursdays at 4:00 in room 2-285. Late problem sets are not accepted.

Problem set 1 due September 13th. Solutions.
Problem set 2 due September 20th. Solutions.
Problem set 3 due September 27th. Solutions.
Review sheet for Midterm 1. Solutions to the midterm.
Problem set 4 due October 11th. Solutions.
Problem set 5 due October 18th. Solutions.
Problem set 6 due October 25th. Solutions.
Problem set 7 due November 1st. Solutions.
Review sheet for Midterm 2. Solutions to the midterm.
Problem set 8 due November 15th. Solutions.
Problem set 9 due November 29th. Solutions.
Problem set 10 due December 6th. Solutions.
Review sheet for the final.

Office hours for the final:

Wednesday 12/12, 11:00-12:00 in 2-173 (Kiril Datchev)
Wednesday 12/12, 5:00-7:00 in 2-492 (David Jackson-Hanen, 18.100B/C teaching assistant)
Thursday 12/13, 5:00-7:00 in 2-492 (David Jackson-Hanen, 18.100B/C teaching assistant)
Monday 12/17, 2:00-4:00 in 2-270 (Paul Seidel, 18.100C professor)
Monday 12/17, 5:00-7:00 in 2-492 (David Jackson-Hanen, 18.100B/C teaching assistant)
Tuesday 12/18, 12:00-1:00 in 2-173 (Kiril Datchev)
Tuesday 12/18, 2:00-4:00 in 2-270 (Paul Seidel, 18.100C professor)
Tuesday 12/18, 5:00-7:00 in 2-492 (David Jackson-Hanen, 18.100B/C teaching assistant)
Wednesday 12/19, 12:00-1:00 in 2-173 (Kiril Datchev)
More office hours to be announced here.

Schedule

Date Pages Topics

9/6 1 – 12 ordered sets, fields, real numbers

9/11 12 – 26 complex numbers, Euclidean spaces, functions, finite and infinite sets

9/13 26 – 34 countable and uncountable sets, metric spaces

9/18 34 – 36 open and closed sets

9/20 36 – 38 compact sets

9/25 38 – 40 the Heine-Borel theorem, the Bolzano-Weierstrass theorem

9/27 40 – 43 perfect sets, the Cantor set, connected sets

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

10/4 47 – 54 sequences, convergence, Cauchy sequences

10/11 54 – 60 completeness, monotonic sequences, upper and lower limits, series, comparison test

10/16 60 – 65 series of nonnegative terms, the number e

10/18 65 – 72 root and ratio tests, power series, conditional and absolute convergence

10/23 83 – 90 continuous functions, continuity and compactness

10/25 90 – 97 uniform continuity, continuity and connectedness, monotonic functions

10/30 103 – 109 differentiation, mean value theorems

11/1 109 – 113 l'Hospital's rule, Taylor's theorem

11/6 47 – 119 Midterm on Chapters 3, 4 and 5

11/8 120 – 127 the Riemann-Stieltjes integral

11/13 128 – 137 properties of the integral, fundamental theorem of calculus, rectifiable curves

11/15 140 – 148 sequences and series of functions, pointwise and uniform convergence

11/20 148 – 154 uniform convergence, continuity and differentiation

11/27 154 – 158 equicontinuous families of functions, the Arzelà-Ascoli theorem

11/29 159 – 165 the Stone-Weierstrass theorem

12/4 172 – 178 functions given by power series

12/6 178 – 184 exponential, logarithmic and trigonometric functions

12/11 185 – 192 Fourier series

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

Schedule
Date	Pages	Topics
9/6	1 – 12	ordered sets, fields, real numbers
9/11	12 – 26	complex numbers, Euclidean spaces, functions, finite and infinite sets
9/13	26 – 34	countable and uncountable sets, metric spaces
9/18	34 – 36	open and closed sets
9/20	36 – 38	compact sets
9/25	38 – 40	the Heine-Borel theorem, the Bolzano-Weierstrass theorem
9/27	40 – 43	perfect sets, the Cantor set, connected sets
10/2	1 – 46	Midterm on Chapters 1 and 2
10/4	47 – 54	sequences, convergence, Cauchy sequences
10/11	54 – 60	completeness, monotonic sequences, upper and lower limits, series, comparison test
10/16	60 – 65	series of nonnegative terms, the number e
10/18	65 – 72	root and ratio tests, power series, conditional and absolute convergence
10/23	83 – 90	continuous functions, continuity and compactness
10/25	90 – 97	uniform continuity, continuity and connectedness, monotonic functions
10/30	103 – 109	differentiation, mean value theorems
11/1	109 – 113	l'Hospital's rule, Taylor's theorem
11/6	47 – 119	Midterm on Chapters 3, 4 and 5
11/8	120 – 127	the Riemann-Stieltjes integral
11/13	128 – 137	properties of the integral, fundamental theorem of calculus, rectifiable curves
11/15	140 – 148	sequences and series of functions, pointwise and uniform convergence
11/20	148 – 154	uniform convergence, continuity and differentiation
11/27	154 – 158	equicontinuous families of functions, the Arzelà-Ascoli theorem
11/29	159 – 165	the Stone-Weierstrass theorem
12/4	172 – 178	functions given by power series
12/6	178 – 184	exponential, logarithmic and trigonometric functions
12/11	185 – 192	Fourier series
12/20	1 – 192	Final Exam on Chapters 1 – 8