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
 Sigmaalgebras
 Measures, Outer Measures
 Borel Measures on the Real Line, Nonmeasurable Sets

Integration
 Measurable Functions
 Littlewood's Three Principles
 Integration of Nonnegative Functions
 Integration of Complex Functions
 Modes of Convergence
 Product Measures
 The Ndimensional Lebesgue Integral
 Integration in Polar Coordinates

Signed Measures and Differentiation
 The HardyLittlewood Maximal Function
 Signed Measures
 The LebesgueRadonNikodym Theorem
 Complex Measures
 Differentiation on Euclidean Space
 Functions of Bounded Variation

$L^p$ Spaces
 Chebyshev, CauchySchwartz, 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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