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:

* 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.

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