This is a 23-lecture course, with each lecture being about 60-90 minutes, given at MIT in person by Casey Rodriguez. It gives an introduction to functional analysis.
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.
- Basic Banach Space Theory
- Bounded Linear Operators
- Quotient Spaces, the Baire Category Theorem and the Uniform Boundedness Theorem
- The Open Mapping Theorem and the Closed Graph Theorem
- Zorn’s Lemma and the Hahn-Banach Theorem
- The Double Dual and the Outer Measure of a Subset of Real Numbers
- σ-algebras
- Lebesgue Measurable Subsets and Measure
- Lebesgue Measurable Functions
- Simple Functions
- The Lebesgue Integral of a Non-negative Function and Convergence Theorems
- Lebesgue Integrable Functions, the Lebesgue Integral and the Dominated Convergence Theorem
- Lp Space Theory
- Basic Hilbert Space Theory
- Orthonormal Bases and Fourier Series
- Fejer’s Theorem and Convergence of Fourier Series
- Minimizers, Orthogonal Complements and the Riesz Representation Theorem
- The Adjoint of a Bounded Linear Operator on a Hilbert Space
- Compact Subsets of a Hilbert Space and Finite-Rank Operators
- Compact Operators and the Spectrum of a Bounded Linear Operator on a Hilbert Space
- The Spectrum of Self-Adjoint Operators and the Eigenspaces of Compact Self-Adjoint Operators
- The Spectral Theorem for a Compact Self-Adjoint Operator
- The Dirichlet Problem on an Interval
These videos are of a lecture course by Casey Rodriguez at the Massachusetts Institute of Technology in 2021, and made available as part of its OpenCourseWare initiative.
