# 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:

* 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.

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
• $L^p$ Spaces
• Distribution Functions and Weak $L^p$
• Interpolation of $L^p$ Spaces