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.

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