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| MAT4014 | Introduction to Functional Analysis | 4+0+0 | ECTS:6 | | Year / Semester | Spring Semester | | Level of Course | First Cycle | | Status | Elective | | Department | DEPARTMENT of MATHEMATICS | | Prerequisites and co-requisites | None | | Mode of Delivery | | | Contact Hours | 14 weeks - 4 hours of lectures per week | | Lecturer | Prof. Dr. Zameddin İSMAİLOV | | Co-Lecturer | As. Prof. Pembe İpek Al | | Language of instruction | Turkish | | Professional practise ( internship ) | None | | | | The aim of the course: | | To present the basics of modern functional analysis, introducing the lineer normed spaces and transformations on these spaces.
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| Learning Outcomes | CTPO | TOA | | Upon successful completion of the course, the students will be able to : | | | | LO - 1 : | calculate the Fourier coefficients of certain elementary functions. | 5,6 | 1 | | LO - 2 : | perform a range of calculations involving orthogonal expansions in Hilbert spaces and to prove the standard theorems associated with them. | 5,6 | 1 | | LO - 3 : | apply functional analytic techniques to the study of Fourier series. | 5,6 | 1 | | LO - 4 : | give the definitions and basic properties of various classes of operators on a Hilbert space and use them in specific examples. | 5,6 | 1 | | LO - 5 : | prove results related to the theorems in the course. | 5,6 | 1 | | 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), LO : Learning Outcome | | |
| Metric spaces ; Connectedness, Completeness,and compactness; Normed lineer spaces and Banach spaces; Inner product spaces, Hilbert spaces and orthogonal expansions; Lineer bounded functionals, dual spaces and Hahn-Banach Theorem; Lineer bounded operators; Lineer bounded operators in Hilbert spaces; ; Spectrum of an operator ; Compact operators.
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| Course Syllabus | | Week | Subject | Related Notes / Files | | Week 1 | Metric spaces ; Convergence , open and closed sets in metric spaces; Limit and continuity in metric spaces | | | Week 2 | Connected, complete and compact metric spaces; Continuous functions in metric spaces | | | Week 3 | Banach fixed point theorem and its applications | | | Week 4 | Linear normed and seminormed spaces | | | Week 5 | Banach spaces | | | Week 6 | Inner product spaces and Hilbert spaces | | | Week 7 | Orthogonal systems and orthogonal decompositions in inner product spaces | | | Week 8 | Linear bounded functionals and Riesz-Frechet's Theorem | | | Week 9 | Mid-term exam | | | Week 10 | Dual spaces, Hahn-Banach Theorem
| | | Week 11 | Linear bounded operators | | | Week 12 | Linear bounded operators in Hilbert spaces | | | Week 13 | Spectrum of an operator | | | Week 14 | Linear compact operators | | | Week 15 | Applications | | | Week 16 | Final exam | | | |
| 1 | B. Musayev, M.Alp, Fonksiyonel Analiz, Kütahya, 2008 | | | |
| 1 | B.V.Limaye, Linear Functional Analysis for Scientists and Engineers, Springer, 2016 | | | 2 | Bryan P. Rynne and Martin A Younson,Linear Functional Analysis, Second Edition Springer-Verlag,2008. | | | |
| Method of Assessment | | Type of assessment | Week No | Date | Duration (hours) | Weight (%) | | Mid-term exam | 9. hafta | 15.04.2025 | 2 | 50 | | End-of-term exam | 16.hafta | 03.06.2025 | 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 | 4 | 14 | 56 | | Sınıf dışı çalışma | 4 | 8 | 32 | | Arasınav için hazırlık | 7 | 5 | 35 | | Arasınav | 2 | 1 | 2 | | Dönem sonu sınavı için hazırlık | 7 | 7 | 49 | | Dönem sonu sınavı | 2 | 1 | 2 | | Diğer 1 | 4 | 1 | 4 | | Total work load | | | 180 |
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