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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.
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Description
Topics include normed spaces, completeness, functionals, the Hahn-Banach Theorem, duality, operators; Lebesgue measure, measurable functions, integrability, completeness of Láµ– spaces; Hilbert spaces; compact and self-adjoint operators; and the Spectral Theorem
Course content
- Basic Banach Space Theory Unlimited
- Bounded Linear Operators Unlimited
- Quotient Spaces, the Baire Category Theorem and the Uniform Boundedness Theorem Unlimited
- The Open Mapping Theorem and the Closed Graph Theorem Unlimited
- Zorn’s Lemma and the Hahn-Banach Theorem Unlimited
- The Double Dual and the Outer Measure of a Subset of Real Numbers Unlimited
- Sigma Algebras Unlimited
- Lebesgue Measurable Subsets and Measure Unlimited
- Lebesgue Measurable Functions Unlimited
- Simple Functions Unlimited
- The Lebesgue Integral of a Nonnegative Function and Convergence Theorems Unlimited
- Lebesgue Integrable Functions, the Lebesgue Integral and the Dominated Convergence Theorem Unlimited
- Lp Space Theory Unlimited
- Basic Hilbert Space Theory Unlimited
- Orthonormal Bases and Fourier Series Unlimited
- Fejer’s Theorem and Convergence of Fourier Series Unlimited
- Minimizers, Orthogonal Complements and the Riesz Representation Theorem Unlimited
- The Adjoint of a Bounded Linear Operator on a Hilbert Space Unlimited
- Compact Subsets of a Hilbert Space and Finite-Rank Operators Unlimited
- Compact Operators and the Spectrum of a Bounded Linear Operator on a Hilbert Space Unlimited
- The Spectrum of Self-Adjoint Operators and the Eigenspaces of Compact Self-Adjoint Operators Unlimited
- The Spectral Theorem for a Compact Self-Adjoint Operator Unlimited
- The Dirichlet Problem on an Interval Unlimited
Instructor
Massachusetts Institute of Technology
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