MAT 544 Syllabus Real Analysis I Fall 2011

• August 30th
  Unit 1. Advanced calculus and differential equations.
  Metric spaces, convergence, Cauchy sequences, completeness, the real number system.
  Loomis-Sternberg. 4.1, 4.3, 4.4, 4.7
  Folland. 0.6
  
• September 1st
  Sequential compactness, the Bolzano-Weierstrass theorem and the Heine-Borel theorem.
  Loomis-Sternberg. 4.4, 4.7
  Folland. 0.6
  
• September 6th
  Uniform convergency and continuity, equicontinuity, the Arzela-Ascoli theorem.
  Loomis-Sternberg. 4.5, 4.6
  Folland. 4.6
  
• September 8th
  Banach spaces, linear functionals, the Hahn-Banach theorem.
  Loomis-Sternberg. 4.7
  Folland. 5.1, 5.2
  Problem Set 1 due in lecture.
• September 13th
  Hilbert spaces, orthogonal projection, the Gram-Schmidt process.
  Loomis-Sternberg. 5.1, 5.2
  Folland. 5.5
  
• September 15th
  Differentiability.
  Loomis-Sternberg. 3.5, 3.6, 3.7, 3.8, 3.9
  Problem Set 2 due in lecture.
• September 20th
  Banach's Contraction Mapping Fixed Point Theorem. Newton's method. The implicit and inverse function theorems.
  Loomis-Sternberg. 4.9
  
• September 22nd
  Ordinary differential equations and initial value problems. Picard's theorem on short-time existence and uniqueness.
  Loomis-Sternberg. 6.1, 6.2
  Problem Set 3 due in lecture.
• September 27th
  MIDTERM 1 IN LECTURE
  
• October 4th
  Jordan canonical form, matrix exponentiation.
  
• October 6th
  Solutions of constant-coefficient linear ODEs.
  Loomis-Sternberg. 6.3, 6.4, 6.5
  Problem Set 4 due in lecture.
• October 11th
  Unit 2. Measure theory. Sigma-algebras.
  Folland. 1.1, 1.2
  
• October 13th
  Measures.
  Folland. 1.3
  Problem Set 5 due in lecture.
• October 18th
  Outer measures.
  Folland. 1.4
  
• October 20th
  Borel measures on the real line.
  Folland. 1.5
  Problem Set 6 due in lecture.
• October 25th
  Measurable functions.
  Folland. 2.1
  
• October 27th
  Integration of nonnegative functions.
  Folland. 2.2
  Problem Set 7 due in lecture.
• November 1st
  Integration of complex functions.
  Folland. 2.3
  
• November 3rd
  MIDTERM 2 IN LECTURE
  
• November 8th
  Modes of convergence.
  Folland. 2.4
  
• November 10th
  Product measures.
  Folland. 2.5
  Problem Set 8 due in lecture.
• November 15th
  Lebesgue measure on Euclidean spaces.
  Folland. 2.6
  
• November 17th
  Signed measures.
  Folland. 3.1
  Problem Set 9 due in lecture.
• November 22nd
  The Lebesgue-Radon-Nikodym theorem.
  Folland. 3.2
  
• November 29th
  Complex measures.
  Folland. 3.3
  
• December 1st
  Differentiation of measures on Euclidean spaces.
  Folland. 3.4
  Problem Set 10 due in lecture.
• December 6th
  Review
  
• December 8th
  Review
  

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