Math 6100 - Theory of Functions of a Real Variable - Fall 2023

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Syllabus

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Tentative schedule:

Date	Material covered	HW problems and assignments	Remarks
Aug 29	Preliminaries on sets The Cantor-Schröder-Bernstein Theorem
Aug 31	Countable sets Cantor's diagonal argument
Sep 5	Well ordered sets Comparability of well ordered sets	Homework 1 (tex)	Homework is due Tuesday, September 12, by midnight
Sep 7
Sep 12
Sep 14
Sep 19
Sep 21
Sep 26
Sep 28
Oct 3 (will be rescheduled)
Oct 5 (will be rescheduled)
Oct 10
Oct 12
Oct 17
Oct 24
Oct 26
Oct 31
Nov 2
Nov 7
Nov 9
Nov 14
Nov 16
Nov 28
Nov 30
Dec 5
Dec 7
Friday, Dec 15: 3:00pm	Final Exam

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Potential topics to cover:

Preliminaries on sets
The Cantor-Schröder-Bernstein Theorem
Countable sets
Cantor's diagonal argument
Well ordered sets
Comparability of well ordered sets
The axiom of choice
Preliminaries on Metric spaces
Continuity and completeness
Compactness
The Bolzano-Weierstrass property
The Heine-Borel property
Normed spaces
Banach spaces
Bounded continuous functions
Measurable spaces
Vitali sets
Measurable functions
Pointwise limits
Simple functions
Basic properties of measure spaces
Outer measures
Carathéodory's extension theorem
Lebesgue-Stieltjes measures
The Cantor set
The Cantor function
Regularity of Borel measures
Lusin's theorem
Definition of the integral
Integrable functions
Properties of the integral
The monotone convergence theorem
Fatou's lemma
The dominated convergence theorem
Product measures
Fubini's theorem
Lebesgue measure on Euclidean spaces
Exam
Signed measures
Hahn decomposition
Jordan decomposition
Complex measures
Total variation
Lebesgue's decomposition theorem
The Radon-Nikodym theorem
Polar decomposition for a complex measure
Dual of L1
Topological spaces
Separation axioms
Continuity
Nets
Urysohn's lemma
The Tietze extension theorem
Compact spaces
Tychonoff's theorem
The Banach-Alaoglu theorem
The Arzelà-Ascoli theorem
The Stone-Weierstrass theorem
Stone-Čech compactification
Urysohn's metrization theorem
The Baire category theorem
Cantor space
The Cantor-Bendixson theorem
Polish spaces
Suslin scheme's
Lusin's Separation Theorem
Kuratowski's theorem on standard Borel spaces
Standard probability spaces

Mathematical resources:

Math department events
MathSciNet
Zentralblatt Math
arXiv
Mathematics Stack Exchange
MathOverflow

Advice:

Career advice from Terence Tao.
Advice to a young mathematician from Sir Michael Atiyah, Béla Bollobás, Alain Connes, Dusa McDuff, and Peter Sarnak.
Ten lessons I wish I had been taught, by Gian-Carlo Rota.

Interesting links:

David Hilbert radio address.
An interview with Andrei Kolmogorov.
John von Neumann, a documentary.
My analysis prelim from UCLA.
Oberwolfach photo collection.
Math problems.
Banach-Tarski, from the students at the University of Copenhagen.
History of mathematics.
The Mathematics Genealogy Project.
Recording of Carl Friedrich Gauss.

Jesse Peterson's homepage