Analysis

Syllabus
Dates: 
Topics: 


Advanced Calculus/Ordinary Differential Equations (``ODE'') 

Sept 1 
Review of the real number system 
G1.1 
Sept 3 
Review of the real number system 
G1.2 
Sept 8 
Metric spaces, continuity, uniform convergence 
G1.3 
Sept 10 
Metric spaces, continuity, uniform convergence 
G1.41.5 
Sept 15 
Contraction mapping principle Existence and uniqueness theorems for ODE 
G2.12.2 
Sept 17 
Existence and uniqueness theorems for ODE 
G2.2 
Sept 22 
Global existence theorem for linear ODE 
G2.2 
Sept 24 
Linear transformations, orthogonal projections and matrix exponential 
G2.3 
Sept 29 
Tue. Sept 29 CORRECTION DAY:U Classes follow a Monday schedule. 

Oct 1 
Linear transformations, orthogonal projections and matrix exponential 
G2.3, 3.13.2 
Oct 6 
Linear transformations, orthogonal projections and matrix exponential Linear systems of ODE with constant coefficients 
G3.23.3 
Oct 8 
Linear systems of ODE with constant coefficients Derivatives in R^{n} and in Banach spaces 
G3.33.4 
Oct 13 
Derivatives in R^{n} and in Banach spaces 
G3.5,3.6 
Oct 15 
Derivatives in R^{n} and in Banach spaces Newton's method and the inverse function theorem 
G3.63.7 
Oct 20 
Newton's method and the inverse function theorem 
G3.7 
Oct 22 
The implicit function theorem 
G3.8 
Oct 27 
Midterm 


Measure theory 

Oct 29 
Riemann integral in R^{n} 
G4.1 
Nov 3 
Cantortype sets, dyadic decompositions in R^{n} 
G4.2 
Nov 5 
Measures arising from volume functions on open sets 
G4.34.4 
Nov 10 
Measures arising from volume functions on open sets 
G4.44.5 
Nov 12 
Basic properties of the Lebesgue measure 
G4.64.7 
Nov 17 
Measurable and integrable functions 
G5.15.2 
Nov 19 
Measurable and integrable functions Convergence theorems for Lebesgue integrals: monotone and dominated convergence theorems and Fatou's lemma 
5.25.3 
Nov 24 
Convergence theorems for Lebesgue integrals: monotone and dominated convergence theorems and Fatou's lemma 
G5.45.6 
Nov 26 
Thanksgiving Break Nov. 2529 Thanksgiving Break â€“ NO CLASSES INSESSION 

Dec 1 
Integration of complex functions 
G5.75.8 
Dec 3 
Criterion for Riemann integrability 
G6.16.2 
Dec 8 
Criterion for Riemann integrability 
G6.36.4 
Dec 10 
Review 

G#.# stands for a chapter in Geller's book.
Daryl Geller, A first graduate
course in real analysis. Part I,
Solutions Custom
Publishing (can be ordered from
Donna McWilliams P143 Math Tower);
Gerald B. Folland Real Analysis: Modern Techniques and Their Applications (2nd edition), March 1999 WileyInterscience
Walter Rudin, Principles of
mathematical analysis,
3^{rd}
ed., McGrawHill, New York 1976;
Walter Rudin, Real and complex analysis,
3^{rd}
ed., McGrawHill, New York 1987;