Notes

Math 202A

• Metric Spaces
• Completion
• Completion, ctd.
• Completion, ctd.
• Continuity and Topology
• Weak and Strong Topologies
• Quotient Topologies
• Urysohn's Lemma
• Banach Spaces
• Tietze's Lemma
• Tychonoff's Theorem
• Tychonoff's Theorem, ctd.
• Tychonoff's Equivalence to Axiom of Choice
• Compact Metric Spaces
• Arzela-Ascoli Theorem
• Arzela-Ascoli Theorem, ctd.
• Final Remarks on Topology, Introduction to Measure Theory
• Introduction to Measure Theory, cont., The Borel Premeasure
• The Borel Premeasure, cont.
• Outer Measures
• Outer Measures, cont., Measure Construction
• Uniqueness Condition for Extensions
• Uniqueness Condition for Extensions, cont.
• Introduction to Lebesgue Integration
• Properties of $\mathcal S$-SIFs
• Properties of $\mathcal S$ and $\mu$-measurable Functions
• Equivalent formulation of measurability
• Almost Uniform Convergence
• Convergence in measure
• Riesz-Weyl Theorem
• Representative Elements of the Completion of $\mathrm{SIF}(\mathcal S, mu)$
• Properties of the Lebesgue Integral
• Lebesgue Dominated Convergence Theorem
• Introduction to Functional Analysis

Math 202B

• Hahn-Banach Extension Theorem
• Quotient Spaces, Addendum: Bounded Operators
• Weak and Weak-* Topologies
• Hahn-Banach Separation, Loose Ends
• Convex sets, Hilbert Spaces
• Hilbert Spaces
• Dual of $L^1$
• Lebesgue and Radon-Nikodym
• Introduction to Lattice-Ordered Groups, Normed Vector Lattices
• Stone-Weierstrass Theorem on Lattices