MAT 532
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:
- Suggested Reading:
* 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.
* Anthony Knapp. Basic/Advanced Real Analysis. Free online at http://www.math.stonybrook.edu/~aknapp/download.html
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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