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

7. Sigma Algebras

8. Lebesgue Measurable Subsets & Measure

9. Lebesgue Measurable Functions

10. Simple Functions

11. The Lebesgue Integral of a Non-Negative Function, & Convergence Theorems

12. Lebesgue Integrable Functions, The Lebesgue Integral, & The Dominated Convergence Theorem

13. Lp Space Theory

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