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Last updated:

December 24, 2022


Unlimited Duration


This course includes:

Unlimited Duration

Badge on Completion

Certificate of completion

Unlimited Duration


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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