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Introduction to Functional Analysis

Mathematics MIT CC BY-NC-SA 4.0 23 lectures

Functional analysis helps us study and solve both linear and nonlinear problems posed on a normed space that is no longer finite-dimensional, a situation that arises very naturally in many concrete problems. Topics include normed spaces, completeness, functionals, the Hahn-Banach Theorem, duality, operators; Lebesgue measure, measurable functions, integrability, completeness of Lᵖ spaces; Hilbert spaces; compact and self-adjoint operators; and the Spectral Theorem.

Syllabus

  1. 1 Lecture 1: Basic Banach Space Theory
  2. 2 Lecture 2: Bounded Linear Operators
  3. 3 Lecture 3: Quotient Spaces, the Baire Category Theorem and the Uniform Boundedness Theorem
  4. 4 Lecture 4: The Open Mapping Theorem and the Closed Graph Theorem
  5. 5 Lecture 5: Zorn’s Lemma and the Hahn-Banach Theorem
  6. 6 Lecture 6: The Double Dual and the Outer Measure of a Subset of Real Numbers
  7. 7 Lecture 7: Sigma Algebras
  8. 8 Lecture 8: Lebesgue Measurable Subsets and Measure
  9. 9 Lecture 9: Lebesgue Measurable Functions
  10. 10 Lecture 10: Simple Functions
  11. 11 Lecture 11: The Lebesgue Integral of a Nonnegative Function and Convergence Theorems
  12. 12 Lecture 12: Lebesgue Integrable Functions, the Lebesgue Integral and the Dominated Convergence Theorem
  13. 13 Lecture 13: Lp Space Theory
  14. 14 Lecture 14: Basic Hilbert Space Theory
  15. 15 Lecture 15: Orthonormal Bases and Fourier Series
  16. 16 Lecture 16: Fejer’s Theorem and Convergence of Fourier Series
  17. 17 Lecture 17: Minimizers, Orthogonal Complements and the Riesz Representation Theorem
  18. 18 Lecture 18: The Adjoint of a Bounded Linear Operator on a Hilbert Space
  19. 19 Lecture 19: Compact Subsets of a Hilbert Space and Finite-Rank Operators
  20. 20 Lecture 20: Compact Operators and the Spectrum of a Bounded Linear Operator on a Hilbert Space
  21. 21 Lecture 21: The Spectrum of Self-Adjoint Operators and the Eigenspaces of Compact Self-Adjoint Operators
  22. 22 Lecture 22: The Spectral Theorem for a Compact Self-Adjoint Operator
  23. 23 Lecture 23: The Dirichlet Problem on an Interval

Course materials