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| MAT5320 | Advanced Functional Analysis | 3+0+0 | ECTS:7.5 | | Year / Semester | Spring Semester | | Level of Course | Second Cycle | | Status | Elective | | Department | DEPARTMENT of MATHEMATICS | | Prerequisites and co-requisites | None | | Mode of Delivery | Face to face, Group study | | Contact Hours | 14 weeks - 3 hours of lectures per week | | Lecturer | Prof. Dr. Bahadır Özgür GÜLER | | Co-Lecturer | None | | Language of instruction | Turkish | | Professional practise ( internship ) | None | | | | The aim of the course: | | To cover the basic ideas ofBanach spaces and Hibert spaces , and to provide the fundamental notions and language for 20th-century modern analysis. |
| Programme Outcomes | CTPO | TOA | | Upon successful completion of the course, the students will be able to : | | | | PO - 1 : | understand the notion of Banach and Hilbert Spaces,
| 1,2,3 | 1,3,6 | | PO - 2 : | apply basic principles to solve equations in infinite dimension,
| 1,2,3 | 1,3,6 | | PO - 3 : | understand the geometry of Hilbert Spaces,
| 1,2,3 | 1,3,6 | | PO - 4 : | operate with generalized Fourier Series.
| 1,2,3 | 1,3,6 | | CTPO : Contribution to programme outcomes, TOA :Type of assessment (1: written exam, 2: Oral exam, 3: Homework assignment, 4: Laboratory exercise/exam, 5: Seminar / presentation, 6: Term paper), PO : Learning Outcome | | |
| An introduction to the theory of Banach and Hilbert spaces through the use of familiar examples seen from a modern perspective. Functional analysis up to the beginnings of the study of linear operators. |
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| Course Syllabus | | Week | Subject | Related Notes / Files | | Week 1 | Infinite dimensional vector spaces.
| | | Week 2 | Linear independence, Algebraic basis.
| | | Week 3 | Normedspaces,Bounded linear operators.
| | | Week 4 | Banach spaces, The space lp, Spaces of bounded operators, Separable spaces.
| | | Week 5 | Korovkin?s theorem,Weierstrass approximation theorem.
| | | Week 6 | Completion of normed space, Operator norm, Dual space.
| | | Week 7 | Banach's fixed point theorem,
| | | Week 8 | Mid-term exam | | | Week 9 | Picard-Lindelöf theorem, Hilbert Spaces.
| | | Week 10 | Parallelogram and Polarization identities,
| | | Week 11 |
Orthogonality, Orthonormal Sets, Gram-Schmidt procedure, Orthogonal basis, | | | Week 12 |
Parseval?s equality. Bessel?s inequality.
| | | Week 13 |
Generalized Fourier series.
| | | Week 14 | Riesz-Fischer theorem,Convex sets.
| | | Week 15 | Orthogonal projections, Frechet-Riesz theorem.
| | | Week 16 | End-of-term exam | | | |
| 1 | Rudin,Walter. 1987;Real and Complex Analysis,McGraw-Hill, New York,the Third edition | | | |
| 1 | Reed ,M. and Simon,Nad B .1972;Methods of Modern Mathematical Physics. 1. Functional Analysis, Academic Press, New York | | | 2 | Lebedev,V.I.1997;An Introduction to Functional Analysis and Computational Mathematics,Birkhauser | | | |
| Method of Assessment | | Type of assessment | Week No | Date | Duration (hours) | Weight (%) | | Mid-term exam | 9 | 10/2024 | 2 | 30 | | Homework/Assignment/Term-paper | 13 | 11/2024 | 1 | 20 | | End-of-term exam | 16 | 01/2025 | 2 | 50 | | |
| Student Work Load and its Distribution | | Type of work | Duration (hours pw) | No of weeks / Number of activity | Hours in total per term | | Yüz yüze eğitim | 3 | 14 | 42 | | Sınıf dışı çalışma | 8 | 14 | 112 | | Arasınav için hazırlık | 10 | 1 | 10 | | Arasınav | 2 | 1 | 2 | | Ödev | 12 | 1 | 12 | | Dönem sonu sınavı için hazırlık | 20 | 1 | 20 | | Dönem sonu sınavı | 2 | 1 | 2 | | Total work load | | | 200 |
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