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:
- 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:
- 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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