Date |
Material covered |
HW problems and
assignments |
Remarks |
Aug 24 |
Preliminaries on sets
The Cantor-Schröder-Bernstein Theorem
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Aug 26 |
Countable sets
Cantor's diagonal argument
Well ordered sets
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Homework 1 (tex) |
Homework is due Friday, September 2, by 6:00pm |
Aug 29 |
Comparability of well ordered sets
The axiom of choice
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Aug 31 |
Preliminaries on Metric spaces
Continuity and completeness
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Sep 2 |
Compactness
The Bolzano-Weierstrass property
The Heine-Borel property
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Homework 2 (tex) |
Homework is due Friday, September 9, by 6:00pm |
Sep 5 |
Normed spaces
Banach spaces
Bounded continuous functions
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Sep 7 |
Measurable spaces
Vitali sets
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Sep 9 |
Measurable functions
Pointwise limits
Simple functions
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Sep 12 |
Basic properties of measure spaces
Outer measures
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Homework 3 (tex) |
Homework is due Monday, September 19, by 6:00pm |
Sep 14 |
Carathéodory's extension theorem
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Sep 16 |
Lebesgue-Stieltjes measures
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Sep 19 |
The Cantor set
The Cantor function
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Sep 21 |
Regularity of Borel measures
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Homework 4 (tex) |
Homework is due Wednesday, September 28, by 6:00pm |
Sep 23 |
Lusin's theorem
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Sep 26 |
Definition of the integral
Integrable functions
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Sep 28 |
Properties of the integral
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Sep 30 |
The monotone convergence theorem
Fatou's lemma
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Oct 3 |
The dominated convergence theorem
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Oct 5 |
Product measures
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Oct 7 |
Fubini's theorem
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Homework 5 (tex) |
Homework is due Monday, October 17, by 6:00pm |
Oct 10 |
Lebesgue measure on Euclidean spaces
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Oct 12 |
Midterm Exam
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Exam
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Solutions
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Oct 17 |
Signed measures
Hahn decomposition
Jordan decomposition
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Oct 19 |
Complex measures
Total variation
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Oct 21 |
Lebesgue's decomposition theorem
The Radon-Nikodym theorem
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Oct 24 |
Polar decomposition for a complex measure
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Oct 26 |
Dual of L1
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Oct 28 |
Topological spaces
Separation axioms
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Oct 31 |
Continuity
Nets
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Nov 2 |
Urysohn's lemma
The Tietze extension theorem
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Homework 6 (tex) |
Homework is due Wednesday, November 9, by 6:00pm |
Nov 4 |
Compact spaces
Tychonoff's theorem
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Nov 7 |
The Banach-Alaoglu theorem
The Arzelà-Ascoli theorem
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Nov 9 |
The Stone-Weierstrass theorem
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Homework 7 (tex) |
Homework is due Wednesday, November 16, by 6:00pm |
Nov 11 |
Stone-Čech compactification
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Nov 14 |
Urysohn's metrization theorem
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Nov 16 |
The Baire category theorem
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Homework 8 (tex) |
Homework is due Wednesday, November 30, by 6:00pm |
Nov 18 |
Cantor space
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Nov 28 |
The Cantor-Bendixson theorem
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Nov 30 |
Polish spaces
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Homework 9 (tex) |
Homework is due Wednesday, December 7, by 6:00pm
No late homework
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Dec 2 |
Suslin scheme's
Lusin's Separation Theorem
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Dec 5 |
Kuratowski's theorem on standard Borel spaces
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Dec 7 |
Standard probability spaces
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Saturday, Dec 17: 3:00pm |
Final Exam
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Exam
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Solutions |
Jan 10 |
The Vitali covering lemma
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Jan 12 |
Lebesgue's differentiation theorem
Lebesgue's density theorem
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Jan 17 |
Derivatives of monotonic functions
Functions of bounded variation
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Jan 19 |
Absolutely continuous functions
Singular functions
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Homework 1 (tex) |
Homework is due Thursday, January 26, by 6:00pm |
Jan 24 |
Hölder's inequality
Minkowski's inequality
Completeness of Lp-space
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Jan 26 |
Duals of Lp space
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Homework 2 (tex) |
Homework is due Thursday, February 2, by 6:00pm |
Jan 31 |
Introduction to locally convex topological vector space
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Feb 2 |
The open mapping theorem
The closed graph theorem
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Homework 3 (tex) |
Homework is due Thursday, February 9, by 6:00pm |
Feb 7 |
The Hahn-Banach theorem
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Feb 9 |
The Hahn-Banach separation theorem
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Homework 4 (tex) |
Homework is due Thursday, February 16, by 6:00pm |
Feb 14 |
The Krein-Milman theorem
Introduction to Hilbert spaces
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Feb 16 |
The Riesz representation theorem
Bessel's inequality and Parseval's identity
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Homework 5 (tex) |
Homework is due Thursday, February 23, by 6:00pm |
Feb 21 |
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Feb 23 |
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Homework 6 (tex) |
Homework is due Thursday, March 2, by 6:00pm |
Feb 28 |
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Mar 2 |
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Midterm is due Friday, March 3, by 6pm |
Mar 14 |
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Mar 16 |
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Mar 21 |
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Mar 23 |
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Mar 28 |
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Mar 30 |
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Apr 4 |
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Apr 6 |
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Apr 11 |
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Apr 13 |
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Apr 18 |
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Apr 20 |
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Wednesday, May 3: 9:00am |
Final Exam
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