MATH 702 Syllabus

Abstract Integration

• $\sigma$-algebras, positive measures, completion of a measure
• Measurable functions, simple functions
• The Lebesgue integral
• Fatou's lemma, monotone and dominated convergence theorems

Measure and Topology

• Outer measures, Caratheodory's construction
• Borel and Radon measures
• The Riesz representation theorem
• Lebesgue measure on ${\mathbb R}^d$ revisited
• Littlewood's principles, theorems of Egoroff and Lusin

Products Spaces

• Product measures, Monotone class lemma
• Theorems of Fubini and Tonelli
• Convolutions
• Distribution functions

$L^p$ Spaces

• Convex functions and Jensen's inequality
• Inequalities of Hölder and Minkowski
• Completeness of the $L^p$-norm
• Approximation of $L^p$ functions by continuous functions

Banach Spaces

• Basic ideas and examples
• Principle of uniform boundedness
• The open mapping and closed graph theorems
• The Hahn-Banach theorem
• Dual spaces, weak convergence

Hilbert Spaces

• Basic ideas and examples
• Basic identities and inequalities, projections
• Bounded linear functionals
• Orthonormal sets, Bessel's inequality
• Orthonormal bases, Parseval's formula
• The trigonometric system

Differentiation

• Complex and signed measured
• Absolute continuity of measures
• The Lebesgue-Radon-Nikodym theorem and applications
• Maximal functions and Lebesgue points
• Lebesgue's density theorem
• The fundamental theorem of calculus for Lebesgue integral

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