MAT 532

Title: Real Analysis I (previously MAT 544)

Description: Ordinary differential equations; Banach and Hilbert spaces; inverse and implicit function theorems; Lebesque measure; general measures and integrals; measurable functions; convergence theorems for integrals.

Offered: Fall

Credits: 3

Textbook:

  • Suggested Reading:

    * Folland, G.B. (1984). Real Analysis, New York, Wiley.

    * Royden, H.L. (1969). Real Analysis, New York, MacMillan

    * Rudin, W. (1986). Real and Complex Analysis, New York, McGraw Hill (3rd edition).

    * Stein, E.M. and Sharkarchi, R. (2005). Real Analysis, Measure Theory,Integration and Hilbert Spaces, Princeton University Press.

    * Wheeden, R. and Zygmund, A. (1977). Measure and Integral, An Introduction to Real Analysis, New York, M. Dekker.

    * Rudin, W. (1976). Principles of Mathematical Analysis, New York. McGraw Hill. (3rd edition).

    * Stein, E.M. and Sharkarchi, R. (2003). Fourier Analysis, Princeton University Press.

    * Anthony Knapp. Basic/Advanced Real Analysis. Free online at http://www.math.stonybrook.edu/~aknapp/download.html

Major Topics Covered: 

  • Measures
    • Sigma-algebras
    • Measures, Outer Measures
    • Borel Measures on the Real Line, Non-measurable Sets
  • Integration
    • Measurable Functions
    • Littlewood's Three Principles
    • Integration of Nonnegative Functions
    • Integration of Complex Functions
    • Modes of Convergence
    • Product Measures
    • The N-dimensional Lebesgue Integral
    • Integration in Polar Coordinates
  • Signed Measures and Differentiation
    • The Hardy-Littlewood Maximal Function
    • Signed Measures
    • The Lebesgue-Radon-Nikodym Theorem
    • Complex Measures
    • Differentiation on Euclidean Space
    • Functions of Bounded Variation
  • $L^p$ Spaces
    • Chebyshev, Cauchy-Schwartz, Holder, Minkowski Inequalities, Duality
    • Integral Operators
    • Distribution Functions and Weak $L^p$
    • Interpolation of $L^p$ Spaces
    • convolution, Young's Inequality

Graduate Bulletin Course Information

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