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Four nested, non-concentric circles representing topological space, metric spaces, normed, and inner product spaces.

Course18.102 / 18.102123 lectures

Introduction to Functional Analysis

Dr. Casey Rodriguez · Massachusetts Institute of Technology

Start with lecture 1 At MIT OpenCourseWare

About

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.

Lectures

Lecture Videos

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

Syllabus and materials

At MIT OpenCourseWare, as the course publishes them.

Credits and licence

Institution
Massachusetts Institute of Technology
Department
Mathematics
Teacher
Dr. Casey Rodriguez
Term
Spring 2021
Level
Undergraduate
Language
English
Subjects
Mathematics
Source
MIT OpenCourseWare
Licence
CC BY-NC-SA 4.0, as MIT OpenCourseWare states it. Non-commercial use only. MIT OpenCourseWare's terms
Attribution
Dr. Casey Rodriguez. Introduction to Functional Analysis. Spring 2021. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu/. License: Creative Commons BY-NC-SA.
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