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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).
638 years, 9 months
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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 Currilcum
- 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
