Date 
Material covered 
HW problems and
assignments 
Remarks 
Aug 24 
Preliminaries on sets
The CantorSchröderBernstein Theorem



Aug 26 
Countable sets
Cantor's diagonal argument
Well ordered sets

Homework 1 (tex) 
Homework is due Friday, September 2, by 6:00pm 
Aug 29 
Comparability of well ordered sets
The axiom of choice



Aug 31 
Preliminaries on Metric spaces
Continuity and completeness



Sep 2 
Compactness
The BolzanoWeierstrass property
The HeineBorel property

Homework 2 (tex) 
Homework is due Friday, September 9, by 6:00pm 
Sep 5 
Normed spaces
Banach spaces
Bounded continuous functions



Sep 7 
Measurable spaces
Vitali sets



Sep 9 
Measurable functions
Pointwise limits
Simple functions



Sep 12 
Basic properties of measure spaces
Outer measures

Homework 3 (tex) 
Homework is due Monday, September 19, by 6:00pm 
Sep 14 
Carathéodory's extension theorem



Sep 16 
LebesgueStieltjes measures



Sep 19 
The Cantor set
The Cantor function



Sep 21 
Regularity of Borel measures

Homework 4 (tex) 
Homework is due Wednesday, September 28, by 6:00pm 
Sep 23 
Lusin's theorem



Sep 26 
Definition of the integral
Integrable functions



Sep 28 
Properties of the integral



Sep 30 
The monotone convergence theorem
Fatou's lemma



Oct 3 
The dominated convergence theorem



Oct 5 
Product measures



Oct 7 
Fubini's theorem

Homework 5 (tex) 
Homework is due Monday, October 17, by 6:00pm 
Oct 10 
Lebesgue measure on Euclidean spaces



Oct 12 
Midterm Exam

Exam

Solutions

Oct 17 
Signed measures
Hahn decomposition
Jordan decomposition



Oct 19 
Complex measures
Total variation



Oct 21 
Lebesgue's decomposition theorem
The RadonNikodym theorem



Oct 24 
Polar decomposition for a complex measure



Oct 26 
Dual of L1



Oct 28 
Topological spaces
Separation axioms



Oct 31 
Continuity
Nets



Nov 2 
Urysohn's lemma
The Tietze extension theorem

Homework 6 (tex) 
Homework is due Wednesday, November 9, by 6:00pm 
Nov 4 
Compact spaces
Tychonoff's theorem



Nov 7 
The BanachAlaoglu theorem
The ArzelàAscoli theorem



Nov 9 
The StoneWeierstrass theorem

Homework 7 (tex) 
Homework is due Wednesday, November 16, by 6:00pm 
Nov 11 
StoneČech compactification



Nov 14 
Urysohn's metrization theorem



Nov 16 
The Baire category theorem

Homework 8 (tex) 
Homework is due Wednesday, November 30, by 6:00pm 
Nov 18 
Cantor space



Nov 28 
The CantorBendixson theorem



Nov 30 
Polish spaces

Homework 9 (tex) 
Homework is due Wednesday, December 7, by 6:00pm
No late homework

Dec 2 
Suslin scheme's
Lusin's Separation Theorem



Dec 5 
Kuratowski's theorem on standard Borel spaces



Dec 7 
Standard probability spaces



Saturday, Dec 17: 3:00pm 
Final Exam

Exam

Solutions 
Jan 10 
The Vitali covering lemma



Jan 12 
Lebesgue's differentiation theorem
Lebesgue's density theorem



Jan 17 
Derivatives of monotonic functions
Functions of bounded variation



Jan 19 
Absolutely continuous functions
Singular functions

Homework 1 (tex) 
Homework is due Thursday, January 26, by 6:00pm 
Jan 24 
Hölder's inequality
Minkowski's inequality
Completeness of Lpspace



Jan 26 
Duals of Lp space

Homework 2 (tex) 
Homework is due Thursday, February 2, by 6:00pm 
Jan 31 
Introduction to locally convex topological vector space



Feb 2 
The open mapping theorem
The closed graph theorem

Homework 3 (tex) 
Homework is due Thursday, February 9, by 6:00pm 
Feb 7 
The HahnBanach theorem



Feb 9 
The HahnBanach separation theorem

Homework 4 (tex) 
Homework is due Thursday, February 16, by 6:00pm 
Feb 14 
The KreinMilman theorem
Introduction to Hilbert spaces



Feb 16 
The Riesz representation theorem
Bessel's inequality and Parseval's identity

Homework 5 (tex) 
Homework is due Thursday, February 23, by 6:00pm 
Feb 21 



Feb 23 

Homework 6 (tex) 
Homework is due Thursday, March 2, by 6:00pm 
Feb 28 



Mar 2 


Midterm is due Friday, March 3, by 6pm 
Mar 14 



Mar 16 



Mar 21 



Mar 23 



Mar 28 



Mar 30 



Apr 4 



Apr 6 



Apr 11 



Apr 13 



Apr 18 



Apr 20 



Wednesday, May 3: 9:00am 
Final Exam


