Real Analysis I MAT 550 Spring Semester

1. Brief discussion of the measure theory
   • Riesz Representation Theorem
   • Tonelli's and Fubini's Theorems
   • The dual of L¹
   • Radon-Nykodim Theorem
   • Lebesgue's Theorem
   • Hahn Decomposition Theorem
2. L^p spaces, convergence in measure, the dual of L^p
3. Fourier series
   • Riemann-Lebesgue lemma
   • Convergence of Fourier series for differentiable functions
   • Parseval's formula
4. Functional analysis
   • Open mapping and closed graph theorems
   • Uniform boundedness principle
   • Hahn-Banach theorem
   • Existence of orthonormal bases for Hilbert spaces
   • Maximal operator controlling sequences of operators between Banach spaces
5. More measure theory
   • Maximal operators controlling almost everywhere convergence
   • The fundamental theorems of calculus for the Lebesgue integral
   • Change of variables of integration
   • Polar coordinates
6. Partial Differential Equations
   • Separation of variables
   • The heat equation
   • Laplace's equation, the fundamental solution
   • The strong maximum principle and the Liouville theorem
   • The mean-value theorem
   • The Poisson kernel
   • Approximate identities and the Weierstrass theorem on approximation by polynomials
   • The wave equation, d'Alembert's solution
7. Additional Topics
   • Introduction to ergodic theory
   • Birkhoff's Ergodic Theorem

References:

• Daryl Geller, A first graduate course in real analysis. Part II,
  Solutions Custom Publishing (can be ordered from the campus bookstore);
• 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;
• Michael Taylor, Partial differential equations,
  Springer Verlag, 1996.