MAT 533, Real Analysis II
Spring 2021
Christopher Bishop
Office: 4-112 Mathematics Building
Phone: (631)-632-8274
Dept. Phone: (631)-632-8290
FAX: (631)-632-7631
Summary:
MAT 533 is a continuation of MAT 532 and is one of the
requried core courses for first year graduate students.
We cover an introduction to functional analysis, the
Riesz represenation theorem, an introuction to Fourier
analysis and distribution theory. If time permits, we will
do a few things about probability theory and fractal
geometry too.
Syllabus:
The
syllabus
contains information about the textbook and grading.
Textbook:
Real Analysis, 2nd edition, by Gerald Folland
Time and place :
Live online lectures MW 2:40-4:00pm,
Zoom lecture link
   
   
Meeting ID: 974 0855 8490, Passcode: textbook author
   
   
For audio only use phone nummber: 646 876 9923 US (New York)
   
   
Find your local number .
Final Exam:
Final exam will be distributed and uploaded via Blackboard. It will be during the scheduled
exam period: Thursday, May 13, 11:15-1:45.
The final exam will have 5 problems. The first problem has five
subparts that asks for examples of various objects from Chapters 4, 5
and 7. For example, give an example of a meager set with full measure.
You do not need to prove your examples are correct. The other four
problems require a proof of some sort. You will have a choice of four
from five possibilities: two from Chapter 8 (Fourier analysis),
two from Chapter 9 (Distributions) and one from Chapter 10 (probability)
You may use the textbook during the exam, but no other sources
(books, people, notes,...).
Please review the following problems from the text. A few of these
(or variations) will appear on the
final exam: 8.8, 8.18, 8.32, 8.34, 8.35, 9.10, 9.11, 9.25, 9.33, 10.14, 10.15.
Office hours (via Zoom):
Prof Bishop: Wed 4-5, and by appointment.
personal Zoom link .
Meeting ID: 625 069 4063
Passcode: 653434
Jordan Rainone (grader): Monday 1-2pm
personal Zoom link .
Lecture notes:
Since we are online, I will try to make
lecture slides that that summarize the important material
from the text book and occassionally add other remarks:
CHAPTER 4 SLIDES - Topology
CHAPTER 4 SLIDES, ANNOTATED The slide plus any
corrections, skteches,... I made in class. Much larger file.
CHAPTER 5 SLIDES - Functional analysis
CHAPTER 7 SLIDES - Radon measures
CHAPTER 8 SLIDES - Fourier analysis
CHAPTER 9 SLIDES - Distributions and Sobolev spaces
CHAPTER 10 SLIDES - Topics in Probability Theory
Prof Varolin's notes on existence and uniqueness for ODE
Slides based on Varolin's ODE notes
Machine learning
This
subdirectory
contains some math paper related to machine learning. The one labeld "SIAMreview" is the most basic and expository. The rest are currently mostly some papers of Charles Fefferman that were recently pointed
out to me. Send me link for other papers that you think are interesting and accessible to
beginners.
Lecture recordings
Class recordings are available on the class Blackboard page.
Lecture recordings
Previous lectures
Tentative lecture schedule
       
Feb 1: Introduction, overview
       
Feb 3: 4.3, 4.4 Nets and compactness
       
Feb 8: 4.7 Arzela-Asccoli, Tychonoff's theorem,
Stone-Weierstrass theorem
       
Feb 10: 5.1 Normed vector spaces
       
Feb 15: 5.2 Linear functionals
       
Feb 17: 5.3 Baire category
       
Feb 22: 5.4 Topological vector spaces
       
Feb 24: 5.5 Hilbert space
       
Mar 1: 5.5 Hilbert space
       
Mar 3: 7.1 Positive linear functionals
       
Mar 8: 7.2 Regularity and approximation
       
Mar 10: 7.3 Dual of C_0
       
Mar 15: 8.1 Prelinaries and notation
       
Mar 17: 8.2 Convolutions
       
Mar 26-27: Midterm (on Chapter 5)
       
Mar 24: 8.3 The Fourier transform
       
Mar 29: 8.3 The Fourier transform
       
Mar 31: 8.4 Summation of Fourier integrals/series
       
Apr 5: 8.5 Pointwise convergence
       
Apr 7: 8.5 Pointwise convergence
       
Apr 12: 8.6 Fourier analysis of measures
       
Apr 14: 8.7 Applications to PDE
       
Apr 19: 9.1 Distributions
       
Apr 21: 9.2 Tempered distributions
       
Apr 26: 9.3 Sobolev spaces
       
Apr 28: Quick intro on ODE
Prof Varolin's notes on ODE
       
May 3: 10.1 Basic concepts of probability
       
May 5: 10.2 Law of large numbers
Problem sets:
Problem sets should be submitted to the grader. Electronic submissions
of LaTexed solutuions is preferable, but scans or photographs of hand-written
solutions are acceptable if clearly legible. If you strongly prefer to
hand in physical solution sets, contact the grader to make arrangments to
drop them at his office, or leave them with the graduate secretary.
My plan to is have problems sets due every other week. My hope was
that this was less stressful than weekly assignments, although the
total number of problems would be the same. Tentatively, problems sets
are due on the Friday following the week(s) when we cover the relevant
section. Due dates may change
depending on the schedules for other core classes, and the grader.
Late problem sets will be accepted, but with a penalty determined
by the grader.
       
Due Friday, Feb 12: 4.36 (you may use cited exercises without proof), 4.42, 4.43, 4.68
       
Due Friday, Feb 26: 5.5, 5.7, 5.12, 5.18, 5.19
       
Due Friday, March 12: 5.27, 5.42, 5.47, 5.53, 5.59, 5.66
       
Friday, March 26: No problem set due. The midterm takes its place
this week. The exam will be made available on Blackboard at
9am on Friday, March 26, 2021 and should be uploaded in Blackboard
(or emailed to the lecturer) by 9pm Saturday, March 27 2021.
       
Due Friday, April 9: 7.17, 7.20, 7.22, 7.24 (7.24 needs a correction. See
Folland's errata page )
       
Due Friday, April 30 (revised from April 23): 8.3, 8.10, 8.11, 8.15, 8.18
       
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.
The not too short introduction to LaTex
Additional links
Hugh Woodin, The Continuum
Hypothesis, Part I
This gives an introduction to set theory with a discussion of the
the role of the axiom of choice and the existence of non-measurable sets.
Hugh Woodin, The Continuum
Hypothesis, Part II
This continues the previous article and discusses in what sense the continuum
hypothesis can be considered true or false, even through it is formally
independent of ZFC.
paper giving careful proof of Banach-Tarski paradox
Wikipedia article on the Banach-Tarski paradox
Wikipedia article on Carleson's a.e. convergence theorem
Wikipedia article on Weierstrass' nowhere differentiable function
Link to
Schroder-Bernstein theorem
Link to
Freilng's dart argument against CH
Link to
history of mathematics
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seek notes from a classmate or identified class note
taker, write lab report based on sample data).
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Send me email at:
bishop at math.sunysb.edu
Link to
history of mathematics