Math 104. Introduction to Analysis

Department of Mathematics
University of California, Berkeley
Spring 2010

This is an introductory course on analysis.

The official prerequisites for taking this course are Math 53: Multivariable Calculus and Math 54: Linear Algebra and Differential Equations.

Announcements

05/13/10: Office hours 3:30–4:30 today.
05/08/10: Office hours 2:30–3:30 next Mon.
05/04/10: Office hours 2–3 this Wed and 3–4 this Fri.
04/30/10: Final exam syllabus posted.
04/30/10: Next Mon office hours 12–1.
04/30/10: Homework 9 posted.
04/25/10: This Wed office hours moved to 3–4.
04/23/10: Homework 8 posted.
04/16/10: Homework 7 posted.
04/08/10: Final exam in Valley LSB, Room 2040, 8–11am, May 14 (Fri) (note: venue given in official final exam schedule is incorrect).
04/08/10: Homework 6 posted.
04/06/10: Midterm II in GSPP, Room 150, 4–5, April 12 (Mon).
03/31/10: Homework 5 posted.
03/29/10: This Wed office hours moved to 3–4.
03/21/10: Homework 4 posted.
03/08/10: Homework 3 posted.
03/03/10: Midterm I in Leconte Hall, Room 2, 4–5, March 12 (Fri).
02/22/10: Midterm I on March 12 (Fri), 4–5; Midterm II on April 12 (Mon), 4–5.
02/22/10: Homework 2 posted.
02/10/10: Homework 1 posted.
01/23/10: As discussed, the class will meet for an extra 30 minutes (4–5:30) from now on to make up for the cancelled lectures.
01/14/10: No lecture on Wed, Jan 20. First lecture will take place on Fri, Jan 22.
01/14/10: Check this page regularly for announcements.

Lectures

Location: Evans Hall, Room 9

Times: 4:00–5:30 PM on Mon/Wed/Fri

Course staff

Instructor: Lek-Heng Lim
Evans Hall, Room 873
lekheng(at)math.berkeley.edu
(510) 642-8576
Office hours: 3–4, Mon; 12–1, Wed; 3–4, Fri.

Graduate Student Instructor: Aaron McMillan
Evans Hall, Room 891
aaronfm(at)math.berkeley.edu
Office hours: 9–12 & 4–6, Mon; 9–12 & 1–3, Tue.

Syllabus

Real number system
Sequences
Limits
Continuous functions on R and Rⁿ
Metric space
Uniform convergence
Interchange of limit operations
Infinite series
Differentiable functions on R and Rⁿ
Mean value theorem
Riemann integral

Homework assignments

Homework will be assigned once a week and will be due the following week (except possibly in the weeks when there is a midterm). Collaborations are permitted but you will need to write up your own solutions.

Homework 9: Do exercises 2.1, 2.13, 3.12, 3.18, 3.23, 3.30, 3.37, 3.38, 5.3, 5.4, 5.10, 5.30, 6.2, 6.10, 6.11 (posted: Apr 30; due: May 7); Solutions: see pp. 330–352 in the textbook.

Homework 8: Do exercises 6.9, 6.12, 6.15, 6.20, 6.22, 6.23, 6.25, 6.27 (posted: Apr 23; due: Apr 30); Solutions: PDF (posted: May 6)

Homework 7: Do exercises 5.31, 5.32, 5.33, 4.46, 4.48, 4.50, 4.54, 6.5, 6.6 (posted: Apr 16; due: Apr 23); Solutions: PDF (posted: May 4)

Midterm II (posted: Apr 12); Solutions: PDF (posted: May 8)

Homework 6: Do exercises 5.2, 5.5, 5.6, 5.7, 5.9, 5.11, 5.12, 5.14, 5.16, 5.19 (posted: Apr 8; due: Apr 16); Solutions: PDF (posted: May 4)

Homework 5: Do exercises 4.4, 4.6, 4.8, 4.9, 4.11, 4.13, 4.14, 4.17, 4.18, 4.29, 4.30, 4.31, 4.33, 4.37, 4.40, 4.42 (posted: Mar 31; due: Apr 7); Solutions: PDF (posted: Apr 8)

Homework 4: Do exercises 3.22, 3.24, 3.26, 3.27, 3.29, 3.31, 3.32, 3.33, 3.40, 3.42, 3.43, 3.57 (posted: Mar 21; due: Mar 31); Solutions: PDF (posted: Apr 8)

Midterm I (posted: Mar 12); Solutions: PDF (posted: Mar 29)

Homework 3: Do exercises 3.1, 3.2, 3.3, 3.4, 3.6, 3.8, 3.16, 3.17, 3.19, 3.21 (posted: Mar 8; due: Mar 17); Solutions: PDF (posted: Mar 24)

Homework 2: Do exercises 2.2, 2.4, 2.6, 2.10, 2.11, 2.12, 2.14, 2.16, 2.17, 2.19 (posted: Feb 22; due: Mar 3); Solutions: PDF (posted: Mar 4)

Homework 1: Do exercises 1.5, 1.16, 1.20, 1.25, 1.28, 1.33, 1.37, 1.38, 1.43 (posted: Feb 10; due: Feb 19); Solutions: PDF (posted: Mar 6)

Bug report on the problem sets or the solutions: lekheng(at)math.berkeley.edu

Supplementary materials

Course homepage from Spring 2009.

Some notes on sets, logic, and mathematical language (courtesy of George Bergman).

Grades

Grade composition: best 8 of 10 homeworks (1st midterm counts as a homework) [40%], 2nd midterm [15%], final exam [45%].

Statistics: 1st midterm (mean = 8.25/12, median = 8.5/12), 2nd midterm (mean = 8/14, median = 8.25/14).

Final exam syllabus

General rule of thumb: If it's not covered in the lectures, it won't be on the final. In the following, "up to" means "up to and including".

Chapter 1: Definition 1.12, Section 1.5.
Chapter 2: Section 2.1, Section 2.2, Section 2.3 up to statement of Theorem 2.12.
Chapter 3: Section 3.1, Section 3.2 up to Example 3.14, Section 3.3, Definition 3.9.
Chapter 4: Section 4.1, Section 4.2 up to Theorem 4.7, Section 4.3 up to Theorem 4.13.
Chapter 5: Section 5.1, Section 5.2 up to Theorem 5.7, Section 5.3 up to Theorem 5.12.
Chapter 6: Section 6.1, Section 6.2, Section 6.3.