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:
- 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
- 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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