MAT 533

Title: Real Analysis II (formerly MAT 550)

Description: Representations and decomposition theorems in measure theory; Fubini's theorem; L-p spaces; Fourier series; Laplace, heat and wave equations; open mapping and uniform boundedness theorems for Banach spaces; differentation of the integral; change of variable of integration.

Offered: Spring

Prerequisite: MAT 544

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: 

  • Compactness
    • Arzela-Ascoli, Stone-Weierstrass
  • Functional Analysis
    • Normed Vector Spaces
    • Linear Functionals, Hahn-Banach Theorem
    • Baire Category Theorem, Open Mapping Theorem, Closed Graph Theorem, Uniform Boundedness Principle
    • Topological Vector Spaces, Duality, Weak and Weak* Convergence, Alaoglu's Theorem
    • Hilbert Spaces
  • $L^p$ Spaces
    (completing Only What Was Omitted in First Semester)
  • Ordinary Differential Equations
  • Radon Measures on Locally Compact Hausdorff Spaces
  • Elements of Fourier Analysis
    • Fourier Transform on $R^n$ and the Circle
    • Riemann Lebesgue Lemma, Hausdorff-Young Inequality, Plancharel, Poisson Summation, $L^2(R^n)$
    • Summation and Convergence Theorems
  • Distributions

Graduate Bulletin Course Information

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