MAT 550, Real Analysis II

Spring 2012

Christopher Bishop

Professor, Mathematics
SUNY Stony Brook

Office: 4-112 Mathematics Building
Phone: (631)-632-8274
Dept. Phone: (631)-632-8290
FAX: (631)-632-7631

Time and place: TuTh 9:50-11:10, Math 4-130

We will use the text `Real Analysis: modern techniques and their applications' by Gerald Foland, second edition.

This course is a continuation of MAT 544 in Fall 2011. I expect to cover Chapter 3, selected sections of Chapters 4 and 5 and most of Chapters 6,7,8 and 9. In time permits, I will try to cover some sections from chapters 10 and 11 and perhaps add some supplementary material on PDEs.

My office hours will be Tu-Th 11:10-12:00 in my office, 4-112 in the Math Building.

Homework problems will be asssigned from the text each week and will be handed in at class each Thursday.

Although it is not required, you may wish to consider writing up your solution in TeX, since eventually you will probably use this to write your thesis and papers. Here are a sample LaTex file and what the resulting output looks like . You can use the first file as a template to create your own TeX files. Numerous guides exist online that give the basic rules and commands.

The first lecture is Tuesday, Jan 24. The last class meeting is Thursday May 3. There is no class during spring recess: April 2-8. There will be a midterm exam on Thursday March 29 (date tentative; location in Math 4-130) and a final at 11:15am-1:45pm on Thursday, May 10 (regular room, Math 4-130). Homework, midterm and final will each count for one third of your grade.

Lecture Schedule (this is tentative and only approximate; it will almost certainly change as we proceed).

Link to Sample MIDTERM

Lecture 1, Tuesday, Jan 24
Overview of course

Lecture 2, Thursday, Jan 26
3.1 Signed Measures
Homework: 3,4,5,7 Due Thursday Feb 2

Lecture 3, Tuesday, Jan 31
3.2 Radon-Nikodym
Homework: 10,11,14,17 Due Thursday Feb 9

Lecture 4, Thursday, Feb 2
3.3 Complex measures

Lecture 5, Tuesday, Feb 7
3.4 Lebesgue Differentiation
Homework: 22,23,24,25 Due Thursday Feb 16

Lecture 6, Thursday Feb 9
3.5 Bounded variation
Homework: 30, 31, 37, 39, 40

Lecture 7, Tuesday, Feb 14
3.5 continued

Lecture 8, Thursday, Feb 16
Review of 4.7, 5.1, 5.2: Stone-Weierstrass, normed vector spaces, Hahn-Banach
Homework:

Lecture 9, Tuesday, Feb 21
5.3 Baire Category; open mapping, closed graph, uniform boundedness
Homework: 27,33,40,41,42, due March 8

Lecture 10, Thursday, Feb 23
6.1 L^p spaces
Homework: 5,7, 10, 12, 18, 19, Due March 8

Lecture 11, Tuesday, Feb 28
6.2 Dual Spaces 6.4 Weak L^p
Homework: 27,30,36,38,40

Lecture 12, Thursday, Mar 1
6.3 Some inequalities

Lecture 13, Tuesday, Mar 6
Class Cancelled

Lecture 14, Thursday, Mar 8
6.3 continued, 6.4 Weak L^p

Lecture 15, Tuesday, Mar 13
6.4 Interpolation
Homework: 37, 40, 42, 43, 45

Lecture 16, Thursday, Mar 15
6.4 continued Introduction to Fourier Analysis
Homework:

Lecture 17, Tuesday, Mar 20
6.4 continued (short class today)
Homework:

Lecture 18, Thursday, Mar 22
8.1 Introduction to Fourier analysis
Homework:

Lecture 19, Tuesday, Mar 27
REVIEW
Homework:

Lecture 20, Thursday, Mar 29
MIDTERM
Homework:

April 2-9: SPRING BREAK

Lecture 21, Tuesday, Apr 10
8.2 Convolutions ,

Lecture 22, Thursday, Apr 12
8.2 continued

Lecture 23, Tuesday, Apr 17
8.3 Fourier series

Lecture 24, Thursday, Apr 19
8.3 Fourier transform
Last Homework: 8,9,10, 14,18, 25

Lecture 25, Tuesday, Apr 17
8.4
Homework:

Lecture 26, Thursday, Apr 19
8.5
Homework:

Lecture 27, Tuesday, Apr 24
9.1
Homework:

Lecture 28, Thursday, Apr 26
9.2
Homework:

Lecture 29, Tuesday, May 1
9.3
Homework:

Lecture 30, Thursday, May 3

Thursday, May 10, FINAL, 11:15-1:45, Room TBA

Send me email at: bishop at math.sunysb.edu

Link to history of mathematics