Introduction to
Functional Analysis
Dr. Casey Rodriguez - MIT
1. Basic Banach Space Theory
2. Bounded Linear Operators
3. Quotient Spaces, The Baire Category Theorem, & The Uniform Boundedness Theorem
4. The Open Mapping Theorem, & The Closed Graph Theorem
5. Zorn's Lemma, & The Hahn-Banach Theorem
6. The Double Dual, & The Outer Measure of a Subset of Real Numbers
8. Lebesgue Measurable Subsets & Measure
9. Lebesgue Measurable Functions
11. The Lebesgue Integral of a Non-Negative Function, & Convergence Theorems
12. Lebesgue Integrable Functions, The Lebesgue Integral, & The Dominated Convergence Theorem
14. Basic Hilbert Space Theory
15. Orthnomal Bases & Fourier Series
16. Fejer's Theorem & Convergence of Fourier Series
17. Minimisers, Orthogonal Complements, & Reisz Representation Theorem
18. The Adjoint of a Bounded Linear Operator on a Hilbert Space
19. Compact Subsets of a Hilbert Space & Finite-Rank Operators
20. Compact Operators & The Spectrum of a Bounded Linear Operator on a Hilbert Space
21. The Spectrum of Self-Adjoint Operators & Eigenspaces of Compact Self-Adjoint Operators
22. The Spectral Theorem for a Compact Self-Adjoint Operator
23. The Dirichlet Problem of an Interval