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This is an undergraduate course. In this course, students will have an idea about functional analysis. The intention of this course is to cover normed spaces, completeness, functional, Hahn-Banach theorem, duality, and operators. In addition to that, it will also discuss the Lebesgue measure, measurable functions, integrability, completeness of L-p spaces, compact, Hilbert-Schmidt and trace class operators as well as a spectral theorem in this course.

Assessment

This course does not involve any written exams. Students need to answer 5 assignment questions to complete the course, the answers will be in the form of written work in pdf or word. Students can write the answers in their own time. Each answer needs to be 200 words (1 Page). Once the answers are submitted, the tutor will check and assess the work.

Certification

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Course Credit: MIT

Course Curriculum

 Module 01 lec01 Linear spaces, metric spaces, normed spaces 00:20:00 lec02 Linear maps between normed spaces 00:05:00 lec03 Banach spaces 00:25:00 lec04 Lebesgue integrability 00:20:00 lec05 Lebesgue integrable functions form a linear space 00:25:00 lec06 Null functions 00:40:00 lec07 Monotonicity, Fatou’s Lemma and Lebesgue dominated convergence 00:25:00 lec08 Hilbert spaces 00:30:00 lec09 Baire’s theorem and an application 00:10:00 lec10 Bessel’s inequality 00:25:00 lec11 Closed convex sets and minimizing length 00:20:00 lec12 Compact sets. Weak convergence. Weak compactness 00:30:00 lec13 Baire’s theorem. Uniform boundedness. Boundedness of weakly convergent sequences 00:20:00 Module 02 lec14 Fourier series and L2 00:20:00 lec15 Open mapping and closed graph theorems 00:10:00 lec16 Bounded operators. Unitary operators. Finite rank operators 00:20:00 lec17 The second test 00:15:00 lec18 Compact operators 00:20:00 lec19 Fredholm operators 00:25:00 lec20 Completeness of the eigenfunctions 00:20:00 lec21 Dirichlet problem for a real potential on an interval 00:35:00 lec22 Dirichlet problem (cont.) 00:05:00 lec23 Harmonic oscillator 00:20:00 lec24 Completeness of Hermite basis 00:20:00 lec25 The fourier transform on the line 00:20:00 lec26 Hahn-Banach and review 00:20:00 Assessment Submit Your Assignment 00:00:00 Certification 00:00:00

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