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:

   Note: Subject to change - do not buy before confirming with the course instructor

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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