MAT 524 -- Real Variables II 
Course Description

Professor: D. Bradley (http://www.umemat.maine.edu/faculty/bradley/index.html)

This course is a continuation of MAT 523. As such, we will be reinforcing many of the concepts and topics of MAT 523 as well as forging ahead with new material in the study of real analysis, measure theory and functional analysis.

Students who will profit:

Anyone contemplating the pursuit of an advanced degree in a subject with technical requirements in higher mathematics. Most of the theory we now regard as part of modern "real analysis" had its origin in problems arising in applied mathematics, probability, engineering, and theoretical physics.

Text:

Gerald B. Folland, "Real Analysis: Modern Techniques and their Applications," (2nd ed.) John Wiley and Sons, New York, 1999.  Chapters 3, 5, 6, and others as time permits.

References:

• Halsey L. Royden, "Real Analysis," (3rd ed.) MacMillan, New York, 1988.   Chapters 7-10, 13 & 16.
• Paul R. Halmos, "Measure Theory," Springer Graduate Texts in Mathematics, New York, 1950.  Chapters VI, IX & X.
• Walter Rudin, "Real and Complex Analysis," McGraw-Hill, New York, 1987.  Chapters 2, 5-7 & 9.

Syllabus:

• Signed Measures, Complex Measures
• The Hahn/Jordan Decomposition and Radon-Nikodym Theorems
• Functions of Bounded Variation and Absolutely Continuous Functions
• The Fundamental Theorem of Calculus for Lebesgue Integrals
 
• Elements of Functional Analysis
  • Normed Vector Spaces, Linear Functionals
  • The Hahn-Banach Theorem
  • The Baire Category Theorem
  • Closed Graph and Open Mapping Theoresm; Uniform Boundedness Principle
  • Hilbert Spaces
   
• L^p Spaces
• Hölder and Minkowski Inequalities
• Riesz-Fischer Theorem: Completeness of L^p
• Isometries of L^p
• The Dual of Lsupp
• Riesz Representation Theorem
 
• Elements of Fourier Analysis
• Convolutions
• The Fourier Transform
• The Riemann-Lebesgue Lemma
• Fourier Inversion
• Plancherel's Theorem
• The Poisson Summation Formula
• Pointwise Convergence of Fourier Series
• Applications to Partial Differential Equations
 
• Introduction to Probability
• Independence
• Bernstein's proof of the Weierstrass approximation theorem
• Law of Large Numbers
• Central Limit Theorem