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## Last updated:

December 24, 2022

## Duration:

Unlimited Duration

FREE

## This course includes:

Unlimited Duration

## Certificate of completion

Unlimited Duration

### Description

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

### Course Curriculum

• 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

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Massachusetts Institute of Technology

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