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| MAT5110 | Hilbert Spaces | 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, Practical | | Contact Hours | 14 weeks - 3 hours of lectures per week | | Lecturer | Prof. Dr. Zameddin İSMAİLOV | | Co-Lecturer | None | | Language of instruction | Turkish | | Professional practise ( internship ) | None | | | | The aim of the course: | | To present the basics of modern functional analysis; introducing normed lineer spaces, bounded linear operators, Inner product spaces and Hilbert spaces ; and to apply the theory to Fourier analysis. |
| Programme Outcomes | CTPO | TOA | | Upon successful completion of the course, the students will be able to : | | | | PO - 1 : | calculate the Fourier coefficients of certain elementary functions. | 1,2,3,4 | 1,3,6 | | PO - 2 : | perform a range of calculations involving orthogonal expansions in Hilbert spaces and to prove the standard theorems associated with them. | 1,2,3,4 | 1,3,6 | | PO - 3 : | apply functional analytic techniques to the study of Fourier series. | 1,2,3,4 | 1,3,6 | | PO - 4 : | give the definitions and basic properties of various classes of operators on a Hilbert space and use them in specific examples. | 1,2,3,4 | 1,3,6 | | PO - 5 : | prove results related to the theorems in the course. | 1,2,3,4 | 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 | | |
| Linear spaces, Inner product spaces, normed spaces, Banach spaces (basic definitions only) . Hilbert spaces . Subspaces . Linear spans . Orthogonal expansions . Bessel's inequality . The Riesz-Fischer theorem . Orthogonal complements . Fourier series . Fejér's theorem . Parseval's formula . Dual space of a normed space . Self-duality of Hilbert space . Linear operators . B (H) as a Banach space . Adjoints . Hermitian, unitary and normal operators . The spectrum of an operator on a Hilbert space . The spectral radius formula . Compact operators . Hilbert-Schmidt operators . The spectral theorem for compact normal operators . Applications to integral equations |
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| Course Syllabus | | Week | Subject | Related Notes / Files | | Week 1 | Fourier series, basic notions
| | | Week 2 | The vibrating string
| | | Week 3 | Banach Spaces
| | | Week 4 | Inner-product spaces
| | | Week 5 | Completeness
| | | Week 6 | Orthogonality, Bases, Closed Subspaces and Orthogonal Complements,
| | | Week 7 | Bessel's inequality and its mconsequences,
| | | Week 8 | Mid-term exam | | | Week 9 | Fej´er?s Theorem and its consequences,
| | | Week 10 | Subspaces and Orthogonal complements,
| | | Week 11 | Bounded Linear Functionals,
| | | Week 12 | The spectrum of a linear operator,
| | | Week 13 | The adjoint of an operator, | | | Week 14 | Compact Operators,
| | | Week 15 | The spectral theorem for a compact Hermitian operator,
| | | Week 16 | End-of-term exam | | | |
| 1 | Young , Nicholas .1988; An introduction to Hilbert space, Cambridge University Press, Cambridge | | | |
| 1 | Kreyszig, Erwin .1989; Introductory Functional Analysis with Applications, John Wiley and Sons Inc., New York | | | 2 | Rudin, Walter . 1987; Real and Complex Analysis, McGraw-Hill Book Co., New York, third edition | | | 3 | Bollobas, Béla . 1999; Linear Analysis, Cambridge University Press, Cambridge, second edition | | | |
| Method of Assessment | | Type of assessment | Week No | Date | Duration (hours) | Weight (%) | | Mid-term exam | 8 | 23/11/2012 | 2 | 20 | | Homework/Assignment/Term-paper | 15 | 31/12/2012 | 10 | 30 | | End-of-term exam | 16 | 11/01/2013 | 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 | 12 | 96 | | Arasınav için hazırlık | 15 | 1 | 15 | | Arasınav | 2 | 1 | 2 | | Ödev | 16 | 1 | 16 | | Dönem sonu sınavı için hazırlık | 20 | 1 | 20 | | Dönem sonu sınavı | 2 | 1 | 2 | | Diğer 1 | 10 | 1 | 10 | | Diğer 2 | 22 | 1 | 22 | | Total work load | | | 225 |
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