MAT 533
Title: Real Analysis II (formerly MAT 550)
Description: Representations and decomposition theorems in measure theory; Fubini's theorem; Lp 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:
 Real Analysis: Modern Techniques and Their Applications (2nd edition) by Gerald B. Folland
 Suggested Reading:
* Real Analysis (4th edition) by Royden and Fitzpatrick
* Real and Complex Analysis (3rd edition) by Walter Rudin
* Real Analysis, Measure Theory,Integration and Hilbert Spaces by Stein and Sharkarchi
* Measure and Integral, An Introduction to Real Analysis (2nd edition) by Wheeden and Zygmund
* Principles of Mathematical Analysis (3rd edition) by Walter Rudin
* Fourier Analysis: An Introduction by Stein and Sharkarchi
* Basic/Advanced Real Analysis by Anthony Knapp
Major Topics Covered:
 Compactness
 ArzelaAscoli, StoneWeierstrass

Functional Analysis
 Normed Vector Spaces
 Linear Functionals, HahnBanach 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, HausdorffYoung Inequality, Plancharel, Poisson Summation, $L^2(R^n)$
 Summation and Convergence Theorems
 Distributions
Graduate Bulletin Course Information
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