# MAT 533

Title: Real Analysis II (formerly MAT 550)

Description: Representations and decomposition theorems in measure theory; Fubini's theorem; L-p spaces; Fourier series; Laplace, heat and wave equations; open mapping and uniform boundedness theorems for Banach spaces; differentation of the integral; change of variable of integration.

Offered: Spring

Prerequisite: MAT 544

Credits: 3

Textbook:

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

Major Topics Covered:

• Compactness
• Arzela-Ascoli, Stone-Weierstrass
• Functional Analysis
• Normed Vector Spaces
• Linear Functionals, Hahn-Banach Theorem
• Baire Category Theorem, Open Mapping Theorem, Closed Graph Theorem, Uniform Boundedness Principle
• Topological Vector Spaces, Duality, Weak and Weak* Convergence, Alaoglu's Theorem
• Hilbert Spaces
• $L^p$ Spaces
(completing Only What Was Omitted in First Semester)
• Ordinary Differential Equations
• Radon Measures on Locally Compact Hausdorff Spaces
• Elements of Fourier Analysis
• Fourier Transform on $R^n$ and the Circle
• Riemann Lebesgue Lemma, Hausdorff-Young Inequality, Plancharel, Poisson Summation, $L^2(R^n)$
• Summation and Convergence Theorems
• Distributions