Introduction to Functional Analysis(MAT452)
| Course Code | Course Name | Semester | Theory | Practice | Lab | Credit | ECTS |
|---|---|---|---|---|---|---|---|
| MAT452 | Introduction to Functional Analysis | 7 | 3 | 0 | 0 | 3 | 5 |
| Prerequisites | MAT201, MAT261, MAT262 |
| Admission Requirements | MAT201, MAT261, MAT262 |
| Language of Instruction | |
| Course Type | Elective |
| Course Level | Bachelor Degree |
| Course Instructor(s) | Ayşegül ULUS aulus@gsu.edu.tr (Email) |
| Assistant | |
| Objective | The primary objective of the course is to teach, without resorting to measure theory, the fundamental tools of functional analysis, namely metric spaces, normed spaces, Banach spaces, inner product spaces, and Hilbert spaces. In addition, we will address contractive mappings in metric spaces as well as applications of approximation theory in normed and Hilbert spaces. Finally, particular attention will be given to the application of these theories to various mathematical, physical, and economic problems, so that students will have studied and explored their concrete applications. |
| Content |
Metric Spaces: Review Normed Spaces, Banach Spaces Inner Product Spaces, Hilbert Spaces 4 İmportant Theorems about Hilbert Spaces: Projection Theorem, Decomposition Theorem, Riesz Representation Theorem, Hahn-Banach Theorem Applications: Banach Fixed Point Theorem and Approximation Theorems |
| Course Learning Outcomes | The course learning outcome is to be familiar with basic concepts of functional analysis using the basic tools metric spaces, normed spaces and inner product spaces and to be able to understand the 4 important theorems about Hilbert spaces. They will have become familiar with the applications. |
| Teaching and Learning Methods | Course and Recitation Hours |
| References |
Introductory Functional Analysis with Applications, E. Kreyszig, Wiley An İntroduction to Real Analysis, T. Terzioğlu, ODTÜ Fonksiyonel Analizin Yöntemleri, T. Terzioğlu, Matematik Vakfı Fonksiyonel Analiz, E. Şuhubi, İTÜ Vakfı Bir Analizcinin Defterinden Seçtikleri, T.Terzioğlu, NMK Real Analysis with Economic Applications, Efe A. Ök, Princeton University Press |
Theory Topics
| Week | Weekly Contents |
|---|---|
| 1 | Metric Spaces: Review |
| 2 | Further Examples of Metric Spaces: Sequences Spaces, Function Spaces |
| 3 | Completeness |
| 4 | Complete Metric Spaces |
| 5 | Normed Spaces, Banach Spaces |
| 6 | Compactness and Finite Dimension |
| 7 | Linear Operators |
| 8 | Bounded Operators |
| 9 | Linear Functionals |
| 10 | Normed Spaces of Operators and Dual Space |
| 11 | Inner Product Spaces, Hilbert Spaces |
| 12 | 4 İmportant Theorems about Hilbert Spaces: Projection Theorem, Decomposition Theorem, Riesz Representation Theorem, Hahn-Banach Theorem |
| 13 | Application: Banach Fixed Point Theorem |
| 14 | Application: Approximation Theorems |
Practice Topics
| Week | Weekly Contents |
|---|
Contribution to Overall Grade
| Number | Contribution | |
|---|---|---|
| Contribution of in-term studies to overall grade | 2 | 60 |
| Contribution of final exam to overall grade | 1 | 40 |
| Toplam | 3 | 100 |
In-Term Studies
| Number | Contribution | |
|---|---|---|
| Assignments | 0 | 0 |
| Presentation | 0 | 0 |
| Midterm Examinations (including preparation) | 1 | 30 |
| Project | 1 | 30 |
| Laboratory | 0 | 0 |
| Other Applications | 0 | 0 |
| Quiz | 0 | 0 |
| Term Paper/ Project | 0 | 0 |
| Portfolio Study | 0 | 0 |
| Reports | 0 | 0 |
| Learning Diary | 0 | 0 |
| Thesis/ Project | 0 | 0 |
| Seminar | 0 | 0 |
| Other | 0 | 0 |
| Make-up | 0 | 0 |
| Toplam | 2 | 60 |
| No | Program Learning Outcomes | Contribution | ||||
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | ||
| 1 | understands principles of deductive reasoning; has experience to verify well-foundedness and exactness of mathematical statements in systematic ways; | X | ||||
| 2 | can properly state and use concepts and results of major mathematical interest; | X | ||||
| 3 | masters current computational techniques and algorithms; has a good ability in their use; can identify relevant tools, among those one has learned, suitable to solve a problem and is able to judge whether or not one is in possession of these tools; | X | ||||
| 4 | is able to express one’s mathematical ideas in an organised way both in written and oral forms; | X | ||||
| 5 | understands relations connecting substantial concepts and results; can switch from one viewpoint to another on mathematical objects (pictures, formulae, precise statements, heuristic trials, list of examples,...); | X | ||||
| 6 | has followed individually a guided learning strategy; has pursued steps toward the resolution of unfamiliar problems; | X | ||||
| 7 | has a theoretical and practical knowledge in computer science well adapted for learning a programming language; | X | ||||
| 8 | has investigated the relevance of modeling and using mathematical tools in natural sciences and in the professional life; is conscious about historical development of mathematical notions; | X | ||||
| 9 | has followed introduction to some mathematical or non-mathematical disciplines after one’s proper choice; had experience to learn selected subjects according to one’s proper arrangement; | X | ||||
| 10 | masters French language as well as other foreign languages, to a level sufficient to study or work abroad. | X | ||||
| Activities | Number | Period | Total Workload |
|---|---|---|---|
| Class Hours | 1 | 3 | 3 |
| Working Hours out of Class | 14 | 2 | 28 |
| Assignments | 0 | 0 | 0 |
| Presentation | 1 | 3 | 3 |
| Midterm Examinations (including preparation) | 1 | 10 | 10 |
| Project | 1 | 10 | 10 |
| Laboratory | 0 | 0 | 0 |
| Other Applications | 0 | 0 | 0 |
| Final Examinations (including preparation) | 4 | 10 | 40 |
| Quiz | 0 | 0 | 0 |
| Term Paper/ Project | 0 | 0 | 0 |
| Portfolio Study | 0 | 0 | 0 |
| Reports | 1 | 10 | 10 |
| Learning Diary | 0 | 0 | 0 |
| Thesis/ Project | 0 | 0 | 0 |
| Seminar | 0 | 0 | 0 |
| Other | 0 | 0 | 0 |
| Make-up | 0 | 0 | 0 |
| Yıl Sonu | 1 | 10 | 10 |
| Hazırlık Yıl Sonu | 0 | 0 | 0 |
| Hazırlık Bütünleme | 0 | 0 | 0 |
| Total Workload | 114 | ||
| Total Workload / 25 | 4.56 | ||
| Credits ECTS | 5 | ||