Math 140A - Foundations of Analysis     Fall 2008

Instructor:   Wee Teck Gan     wgan@math.ucsd.edu

Lectures: MWF 11 am-11:50am in U413, Room 2

Office Hours: Mon 3-4pm and Wed 2-3pm in AP&M 7240

Teaching Assistant: Ben Wilson     bewilson@math.ucsd.edu.

Section: A01: W 4-4:50 pm, and A02: W 5-5:50 pm in CENTER 201

TA Office Hours: T 1:30-2:30 pm, Th 1-2 pm, F 10-11 am or by appointment, in AP&M 6452.

Text: "Principles of Mathematical Analysis" (third edition) by Walter Rudin, McGraw-Hill.

Syllabus: 140ABC is a rigorous introduction to mathematical analysis. In particular, it gives rigorous development of the foundations necessary for calculus. Most of what we will cover in 140A is in chapters 1-4 of Rudin. The topics are the real and complex number systems, cardinalities, topology of Euclidean space, the notion of metrics and metric spaces, limits, connectedness, completeness and compactness, continuous functions, sequences and series. A less rigorous treatment is given in Math 142AB, but credit cannot be obtained for both 140AB and 142AB.

Prerequisites: MATH 109.

Homework:   Homework is due on Fridays, 3pm, in the TA's dropbox (in AP&M, 6th floor). Late homeworks will not be accepted. It is OK, even encouraged, to discuss homework problems with your classmates, but you are required to write up your own solutions. Please write legibly and in coherent sentences. Each week, 3 of the hoemwork problems will be selected for grading, one of which is the one marked with a * below. The problem marked with a * will not be discussed in sections or during office hours..

HW 1 (Due 10/3): Chap. 1, # 1, 2*, 3, 4, 5

HW 2 (Due 10/10): Chap 1, # 6, 7, 8*, 9, 10, 12, 13, 16, 17, 18, 19

HW 3 (Due 10/17): Chap. 2, # 2, 3, 4, 5*, 6, 7, 8 9

HW 4 (Due 10/24): Chap. 2, # 11, 19, 20*, 22, 23, 24

HW 5 (Due 10/31): Chap. 2, # 12, 13, 15, 16, 25*, 26, 29

HW 6 (Due 11/7): Chap. 2, # 27, 28; Chap 3: # 1, 3, 20*, 23

HW 7 (Due 11/14): Chap. 3, # 4, 5*, 24, 25

HW 8 (Due 11/21): Chap. 3, # 6, 7, 8, 9, 10, 11, 12*

HW 9 (Due 12/5): Chap. 3, # 13, 16; Chap. 4: # 1, 2, 4, 6*, 8, 11, 20, 21

Exams:
Midterm 1:  Fri, Oct 17 in class.    Sample Midterm 1
Midterm 2:  Fri, Nov 14 in class.    Sample Midterm 2
Final: Tues, Dec 9.     Sample Final

Grading: Homework: 20%,   2 midterm exams: 15% each,   final exam: 50%. (The lowest homework score will not be counted. Missed exams will count zero.)

Approximate Lecture Schedule (may be updated during the course):

 wk date Monday Wednesday Friday 0 9/22 Rational Numbers, Incompleteness 1 9/29 Ordered Sets and fields The Real Numbers Dedekind Cuts 2 10/6 Complex Numbers and Euclidean Spaces Infinity: Countable and Uncountable Metric Spaces 3 10/13 Metric Spaces Open/Closed Sets Exam 4 10/20 Open/Closed sets Connected sets Compact Sets 5 10/27 Compact Sets Heine-Borel and Bolzano-Weierstrass Cantor Set 6 11/3 Convergent and Divergent sequences Cauchy sequences and completeness Upper and lower limits 7 11/10 Series e Exam 8 11/17 Tests Power Series Absolute Convergence and Rearrangements 9 11/24 Continuity Uniform Continuity Holiday 10 12/1 Continuity and Compactness Continuity and Connectedness Review 11 11/9 11/9 Exam