Math 414: Analysis II, Spring 2010

General Information

Math 414, Call #16476 (#19323 for grads), Spring 2010
Classes: MWF 12-1 in 311 Addams Hall
Professor: Alex Furman
Office hours: Mondays 2-3 or by appointment
Tutorials: Thursdays 2-3 612 SEO
Contact: furman at math.uic.edu
Text: Understanding Analysis by Stephen Abbot, Springer.

Course Description

Homework and Syllabus

Date Topic Assigned Homework Comments

1/11
1/13
1/15 Reals (review)
Completeness (review)
Cardinality Review
Read 1.3. Ex: 1.3.2, 1.3.6, 1.3.9
Read 1.4, Ex: 1.4.2, 1.4.5, 1.4.7, 1.4.9-10 Notes on the reals
.
HW1: 1.3.2, 1.3.6, 1.3.9 are due

1/18
1/20
1/22 No classes
Cantor's theorem
Cardinalities of more sets Martin Luther King Jr Day
Ex: 1.4.11-12
Read 1.5, Ex: 1.5.3, 1.5.9

1/25
1/27
1/29 Liuoville's numbers
Monotonic functions
Discontinuities for monotonic List 1 of problems on cardinality
.
. HW2: 1.4.11, 1.4.12, 1.5.9 are due
.
List 2

2/1
2/3
2/4
2/5 Cantor set
Class cancelled
Tutorial
Open and closed sets Read 3.1-3.2, Problems: 3.2.1-3
No class
Discussion of hw2 and monotonic, 612 SEO
Read 3.2. Problems: 3.2.4-9 .
furlough day
.
HW3: Problems 4,5,6 from List 2; Ex: 3.2.2

2/8
2/10
2/11
2/12 Closure and Interior
Compact Sets
Tutorial
Compactness Read 3.2; Problems: 3.2.3, 3.2.8-14
Read 3.3
.
. .
.
.
HW4: 3.2.8, 3.2.9, 3.2.12, 3.2.13

2/15
2/17
2/18
2/19 Compactness
Cantor Set
Tutorial
Compactness via open covers Read 3.3
Ex: 3.3.1-4
.
Ex: 3.3.6, 3.3.7

2/22
2/24
2/26 Continuous functions on compact sets
Review
Midterm 1

HW5: Ex: 3.3.5, 3.3.9, 3.4.4, "dist(K,F)>0 problem"
.
Part II (take home)

3/1
3/3
3/5 Uniform continuity
Uniform convergence
The normed space C(K) Read 4.4
.
. .
.
HW5: 4.4.1-3, 4.4.6

3/8
3/10
3/12 Review
~~Arzela-Ascoli~~
Peano curve, Koch's snowflake Study
Read 6.2, Problems: 6.2.1-11
. .
.
.

3/15
3/17
3/19 Uniformly converging series
Differentiating series
Power series .
.
. .
.
.

3/22-27 Spring Break Spring Break Spring Break

3/29
3/31
4/2 Review
Midterm 2
Radius of convergence .
.
. .
.
.

4/5
4/7
4/9 Abel's Theorem, and consequences
.
Review of power series 6.5 pp.171-5 Exs 6.5.1-4, 6.5.8-11
.
.

4/12
4/14
4/16 Branching processes
Riemann Integral
Riemann integrability Read 6.1
Read 7.1, 7.2
Read 7.2 .
Midterm 2 (make up)
.

4/19
4/21
4/23 Integrability criterion
Properties of the integral
Lebesgue theorem Read 7.3
Read 7.4, Problems: 7.4.1, 7.4.4
Read 7.5, 7.6

4/26
4/28
4/30 Lebesgue theorem (cont.)
Improper integrals
Review More on 7.6
.
. HW 6
HW 7
.

Tue 5/4 8-10am, Final Exam

Grading

Announcements

No class on Monday Jan 18 - Martin Luther King Jr. Day.
No class on Wednesday Feb 3 - Furlough day.
Midterm 1 is scheduled for Friday, Feb 26.
Tutorials will be held on Thursdays 2-3 starting Feb 4 in 612 SEO.
Midterm 2 is scheduled for Wed, March 31.

Handouts

Notes on the reals
List 1 - problems on the topic of cardinality.
List 2 - Monotonic functions.
HW 6 - Riemann integral.
HW 7 - Improper integrals.

[Alex Furman] [CV] [Research]

Date	Topic	Assigned Homework	Comments
1/11 1/13 1/15	Reals (review) Completeness (review) Cardinality	Review Read 1.3. Ex: 1.3.2, 1.3.6, 1.3.9 Read 1.4, Ex: 1.4.2, 1.4.5, 1.4.7, 1.4.9-10	Notes on the reals . HW1: 1.3.2, 1.3.6, 1.3.9 are due
1/18 1/20 1/22	No classes Cantor's theorem Cardinalities of more sets	Martin Luther King Jr Day Ex: 1.4.11-12 Read 1.5, Ex: 1.5.3, 1.5.9
1/25 1/27 1/29	Liuoville's numbers Monotonic functions Discontinuities for monotonic	List 1 of problems on cardinality . .	HW2: 1.4.11, 1.4.12, 1.5.9 are due . List 2
2/1 2/3 2/4 2/5	Cantor set Class cancelled Tutorial Open and closed sets	Read 3.1-3.2, Problems: 3.2.1-3 No class Discussion of hw2 and monotonic, 612 SEO Read 3.2. Problems: 3.2.4-9	. furlough day . HW3: Problems 4,5,6 from List 2; Ex: 3.2.2
2/8 2/10 2/11 2/12	Closure and Interior Compact Sets Tutorial Compactness	Read 3.2; Problems: 3.2.3, 3.2.8-14 Read 3.3 . .	. . . HW4: 3.2.8, 3.2.9, 3.2.12, 3.2.13
2/15 2/17 2/18 2/19	Compactness Cantor Set Tutorial Compactness via open covers	Read 3.3 Ex: 3.3.1-4 . Ex: 3.3.6, 3.3.7
2/22 2/24 2/26	Continuous functions on compact sets Review Midterm 1		HW5: Ex: 3.3.5, 3.3.9, 3.4.4, "dist(K,F)>0 problem" . Part II (take home)
3/1 3/3 3/5	Uniform continuity Uniform convergence The normed space C(K)	Read 4.4 . .	. . HW5: 4.4.1-3, 4.4.6
3/8 3/10 3/12	Review ~~Arzela-Ascoli~~ Peano curve, Koch's snowflake	Study Read 6.2, Problems: 6.2.1-11 .	. . .
3/15 3/17 3/19	Uniformly converging series Differentiating series Power series	. . .	. . .
3/22-27	Spring Break	Spring Break	Spring Break
3/29 3/31 4/2	Review Midterm 2 Radius of convergence	. . .	. . .
4/5 4/7 4/9	Abel's Theorem, and consequences . Review of power series	6.5 pp.171-5 Exs 6.5.1-4, 6.5.8-11 . .
4/12 4/14 4/16	Branching processes Riemann Integral Riemann integrability	Read 6.1 Read 7.1, 7.2 Read 7.2	. Midterm 2 (make up) .
4/19 4/21 4/23	Integrability criterion Properties of the integral Lebesgue theorem	Read 7.3 Read 7.4, Problems: 7.4.1, 7.4.4 Read 7.5, 7.6
4/26 4/28 4/30	Lebesgue theorem (cont.) Improper integrals Review	More on 7.6 . .	HW 6 HW 7 .
Tue 5/4	8-10am, Final Exam