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18.102

Introduction to Functional Analysis

Dr. Casey Rodriguez | Spring 2021

Science & Math Mathematics Linear Algebra Mathematical Analysis

前往原始課程

CC BY-NC-SA 4.0

課程簡介

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.

課程資訊

來源	MIT 開放式課程
科系	Mathematics
語言	English
影片數	23

課程影片 (23)

Lecture 1: Basic Banach Space Theory

Lecture 1: Basic Banach Space Theory

Lecture 2: Bounded Linear Operators

Lecture 2: Bounded Linear Operators

Lecture 3: Quotient Spaces, the Baire Category Theorem and the Uniform Boundedness Theorem

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 4: The Open Mapping Theorem and the Closed Graph Theorem

Lecture 5: Zorn’s Lemma and the Hahn-Banach 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 6: The Double Dual and the Outer Measure of a Subset of Real Numbers

Lecture 7: Sigma Algebras

Lecture 7: Sigma Algebras

Lecture 8: Lebesgue Measurable Subsets and Measure

Lecture 8: Lebesgue Measurable Subsets and Measure

Lecture 9: Lebesgue Measurable Functions

Lecture 9: Lebesgue Measurable Functions

Lecture 10: Simple Functions

Lecture 10: Simple Functions

Lecture 11: The Lebesgue Integral of a Nonnegative Function and Convergence Theorems

Lecture 11: The Lebesgue Integral of a Nonnegative Function and Convergence Theorems

Lecture 12: Lebesgue Integrable Functions, the Lebesgue Integral and the Dominated Convergence...

Lecture 12: Lebesgue Integrable Functions, the Lebesgue Integral and the Dominated Convergence...

Lecture 13: Lp Space Theory

Lecture 13: Lp Space Theory

Lecture 14: Basic Hilbert Space Theory

Lecture 14: Basic Hilbert Space Theory

Lecture 15: Orthonormal Bases and Fourier Series

Lecture 15: Orthonormal Bases and Fourier Series

Lecture 16: Fejer’s Theorem and Convergence of Fourier Series

Lecture 16: Fejer’s Theorem and Convergence of Fourier Series

Lecture 17: Minimizers, Orthogonal Complements and the Riesz Representation Theorem

Lecture 17: Minimizers, Orthogonal Complements and the Riesz Representation Theorem

Lecture 18: The Adjoint of a Bounded Linear Operator on a Hilbert Space

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 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 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...

Lecture 21: The Spectrum of Self-Adjoint Operators and the Eigenspaces of Compact Self-Adjoint...

Lecture 22: The Spectral Theorem for a Compact Self-Adjoint Operator

Lecture 22: The Spectral Theorem for a Compact Self-Adjoint Operator

Lecture 23: The Dirichlet Problem on an Interval

Lecture 23: The Dirichlet Problem on an Interval