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Description
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), and 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)
Course Curriculum
- Lecture 01 – Basic Banach Space Theory Unlimited
- Lecture 02 – Bounded Linear Operators Unlimited
- Lecture 03 – Quotient Space, the Baire Category Theorem and the Uniform Boundedness Theorem Unlimited
- Lecture 04 – The Open Mapping Theorem and the Closed Graph Theorem Unlimited
- Lecture 05 – Zorn’s Lemma and the Hahn-Banach Theorem Unlimited
- Lecture 06 – The Double Dual and the Outer Measure of a Subset of Real Numbers Unlimited
- Lecture 07 – Sigma Algebras Unlimited
- Lecture 08 – Lebesgue Measurable Subsets and Measure Unlimited
- Lecture 09 – Lebesgue Measurable Functions Unlimited
- Lecture 10 – Simple Functions Unlimited
- Lecture 11 – The Lebesgue Integral of a Nonnegative Function and Convergence Theorems Unlimited
- Lecture 12 – Lebesgue Integral Functions, the Lebesgue Integral and the Dominated … Unlimited
- Lecture 13 – Lp Space Theory Unlimited
- Lecture 14 – Basic Hilbert Space Theory Unlimited
- Lecture 15 – Orthonormal Bases and Fourier Series Unlimited
- Lecture 16 – Fejer’s Theorem and Convergence of Fourier Series Unlimited
- Lecture 17 – Minimizers, Orthogonal Complements and the Riesz Representation Theorem Unlimited
- Lecture 18 – The Adjoint of a Bounded Linear Operator on a Hilbert Space Unlimited
- Lecture 19 – Compact Subsets of a Hilbert Space and Finite-Rank Operators Unlimited
- Lecture 20 – Compact Operators and the Spectrum of a Bounded Linear Operator on a Hilbert Space Unlimited
- Lecture 21 – The Spectrum of Self-Adjoint Operators and the Eigenspaces of Compact … Unlimited
- Lecture 22 – The Spectral Theorem of a Compact Self-Adjoint Operator Unlimited
- Lecture 23 – The Dirichlet Problem on an Interval Unlimited
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