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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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