MATH 701 Syllabus

Topology of metric spaces I

• Basic notions, examples of metric spaces
• Accumulation points, convergence of sequences
• Completeness
• Compactness and its various formulations
• Connectedness

Topology of metric spaces II

• Continuous maps between metric spaces
• Continuity and compactness, Continuity and connectedness
• The (universal) Cantor set
• Urysohn's lemma, the extension problem
• Baire's category theorem, applications
• The contraction mapping principle, applications

Function spaces

• Sequences of functions, pointwise vs. uniform convergence
• Equicontinuity, the Arzela-Ascoli theorem
• The Stone-Weierstrass theorem
• Most continuous functions are nowhere differentiable
• The existence and uniqueness theorem for ODE's

Lebesgue theory in ${\mathbb R}^n$

• The Riemann integral revisited
• Construction of Lebesgue measure, measurable sets, non-measurable sets
• Measurable functions, basic properties
• The Lebesgue integral
• The monotone convergence theorem
• Fatou's lemma
• The dominated convergence theorem
• Comparison with the Riemann integral

Basic $L^2$ theory

• The inner product structure
• Orthonormal sets and Fourier series
• Completeness of $L^2[0,1]$, the Riesz-Fischer theorem
• Bessel's inequality and Parseval's formula
• Completeness of the trigonometric system in $L^2[0,1]$
• Applications of Fourier series

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