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18.1021

Introduction to Functional Analysis

3-0-9 units · Graduate

Normed spaces, completeness, functionals, Hahn-Banach theorem, duality, operators. Lebesgue measure, measurable functions, integrability, completeness of L-p spaces. Hilbert space. Compact, Hilbert-Schmidt and trace class operators. Spectral theorem. Students in Course 18 must register for the undergraduate version, 18.102 .

Spring 2021

Topics

  • 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
  • Sigma Algebras
  • Lebesgue Measurable Subsets and Measure
  • Lebesgue Measurable Functions
  • Simple Functions
  • The Lebesgue Integral of a Nonnegative Function and Convergence Theorems
  • Lebesgue Integrable Functions, the Lebesgue Integral and the Dominated Convergence Theorem
  • L p 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
  • Final Assignment

Readings

  • There is no assigned textbook for this course. Instead, we will follow lecture notes written by Professor Richard Melrose when he taught the course in 2020, as well as lecture notes taken by an MIT student who took the class with Dr. Rodriguez in 2021.
    Listed reading

Prerequisites & course information

Subject meets with 18.102 Prereq: ( 18.06 , 18.700 , or 18.701 ) and ( 18.100A , 18.100B , 18.100P , or 18.100Q ) G (Spring) 3-0-9 units

Sources