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18.102 - Introduction to Functional Analysis

18.102 Introduction to Functional Analysis (Spring 2021, MIT OCW). Instructor: Dr. Casey Rodriguez. 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. For example, a nonrelativistic quantum particle confined to a region in space can be modeled using a complex valued function (a wave function), an infinite dimensional object (the function's value is required for each of the infinitely many points in the region). Functional analysis yields the mathematically and physically interesting fact that the (time independent) state of the particle can always be described as a (possibly infinite) superposition of elementary wave functions (bound states) that form a discrete set and can be ordered to have increasing energies tending to infinity. The fundamental topics from functional analysis covered in this course 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. (from ocw.mit.edu)

Lecture 19 - Compact Subsets of a Hilbert Space and Finite-Rank Operators

Instructor: Dr. Casey Rodriguez. We show the connection between compact subsets of a Hilbert space and closed, bounded subsets with equi-small tails (a result analogous to the Arzela-Ascoli Theorem). Then, we define finite-rank operators and compact operators.


Go to the Course Home or watch other lectures:

Lecture 01 - Basic Banach Space Theory
Lecture 02 - Bounded Linear Operators
Lecture 03 - Quotient Space, the Baire Category Theorem and the Uniform Boundedness Theorem
Lecture 04 - The Open Mapping Theorem and the Closed Graph Theorem
Lecture 05 - Zorn's Lemma and the Hahn-Banach Theorem
Lecture 06 - The Double Dual and the Outer Measure of a Subset of Real Numbers
Lecture 07 - Sigma Algebras
Lecture 08 - Lebesgue Measurable Subsets and Measure
Lecture 09 - Lebesgue Measurable Functions
Lecture 10 - Simple Functions
Lecture 11 - The Lebesgue Integral of a Nonnegative Function and Convergence Theorems
Lecture 12 - Lebesgue Integral Functions, the Lebesgue Integral and the Dominated Convergence Theorem
Lecture 13 - Lp Space Theory
Lecture 14 - Basic Hilbert Space Theory
Lecture 15 - Orthonormal Bases and Fourier Series
Lecture 16 - Fejer's Theorem and Convergence of Fourier Series
Lecture 17 - Minimizers, Orthogonal Complements and the Riesz Representation Theorem
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 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 Operators
Lecture 22 - The Spectral Theorem of a Compact Self-Adjoint Operator
Lecture 23 - The Dirichlet Problem on an Interval