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.

- 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.

**Location:** Evans Hall,
Room 9

**Times:** 4:00–5:30 PM on Mon/Wed/Fri

**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.

- Real number system
- Sequences
- Limits
- Continuous functions on
**R**and**R**^{n} - Metric space
- Uniform convergence
- Interchange of limit operations
- Infinite series
- Differentiable functions on
**R**and**R**^{n} - Mean value theorem
- Riemann integral

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`

- Course homepage from Spring 2009.

**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).

*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.

- William Parzynski and Philip Zipse,
*Introduction to Mathematical Analysis*, McGraw-Hill, 1983. Buy or Browse

- Tom Körner's notes: 2001, 2007; Martin Hyland's notes

- Richard Earl's notes I: Sequences and Series, Brian Stewart's notes II: Continuity and Differentiability, Terry Lyons's notes III: Integration