Analysis MAT 544 Fall Semester

1. Advanced Calculus and Ordinary Differential Equations (ODE)
   • Review of the real number system
   • Metric spaces, continuity, uniform convergence
   • Contraction mapping principle
   • Existence and uniqueness theorems for ODE
   • Global existence theorem for linear ODE
   • Linear transformations, orthogonal projections and matrix exponential
   • Linear systems of ODE with constant coefficients
   • Derivatives in Rⁿ and in Banach spaces
   • Newton's method and the inverse function theorem
   • The implicit function theorem
2. Measure Theory
   • Riemann integral in Rⁿ
   • Cantor-type sets, dyadic decompositions in Rⁿ
   • Measures arising from volume functions on open sets
   • Basic properties of the Lebesgue measure
   • Measurable and integrable functions
   • Convergence theorems for Lebesgue integrals: monotone and dominated convergence theorems and Fatou's lemma
   • Criterion for Riemann integrability
3. Additional Topics
   • Iterated integrals; Tonelli's and Fubini's Theorems
   • Riesz Representation Theorem
   • Radon-Nikodym Theorem

References:

• Daryl Geller, A first graduate course in real analysis. Part I,
  Solutions Custom Publishing (to be distributed in class)
• Walter Rudin, Principles of mathematical analysis,
  3^rd ed., McGraw-Hill, New York 1976
• Walter Rudin, Real and complex analysis,
  3^rd ed., McGraw-Hill, New York 1987