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Information
| Code | MT408 |
| Name | Functional Analysis |
| Semester | 8. Semester |
| Duration (T+A) | 3-0 (T-A) (17 Week) |
| ECTS | 5 ECTS |
| National Credit | 3 National Credit |
| Teaching Language | Türkçe |
| Level | Lisans Dersi |
| Type | Normal |
| Mode of study | Yüz Yüze Öğretim |
| Catalog Information Coordinator | Dr. Öğr. Üyesi Doğa Can SERTBAŞ |
Course Goal
To grasp the relationship between metric spaces, vector spaces and normed spaces, to aquaint the students with Banach spaces.
Course Content
Metric spaces. Completeness. Vector spaces and norms. Continuous linear maps.
Course Precondition
None.
Resources
Introductory Functional Analysis with Applications, Erwin Kreyszig, John Wiley & Sons Inc., 1978.
Notes
Analiz IV, Ali Nesin, 4. Basım, Nesin Yayıncılık A.Ş., 2010.
Course Learning Outcomes
| Order | Course Learning Outcomes |
|---|---|
| LO01 | Grasps convergence in metric spaces and understands the Cauchy sequence and completeness concepts. |
| LO02 | Grasps the importance of continuity between metric spaces |
| LO03 | Grasps the Banach Fixed Point Theroem |
| LO04 | Grasps the relationship between vector spcaes and normed spaces |
| LO05 | Can relate the basic theorems of analysis with the concepts of normed spaces. |
| LO06 | Grasps the importance of the norm of a linear transformation |
| LO07 | Can define Banach space and give examples. |
Relation with Program Learning Outcome
| Order | Type | Program Learning Outcomes | Level |
|---|---|---|---|
| PLO01 | Bilgi - Kuramsal, Olgusal | Comprehend the ability to prove the mathematical knowledge gained in secondary education on the basis of theoretical basis. | 5 |
| PLO02 | Bilgi - Kuramsal, Olgusal | Understands importance of basic consepts of Algebra, Analaysis and Topology. | 3 |
| PLO03 | Yetkinlikler - Öğrenme Yetkinliği | Mathematical reasoning demonstrates the ability to develop and write mathematical proofs by gaining maturity. | 4 |
| PLO04 | Bilgi - Kuramsal, Olgusal | Demonstrate the ability to express the basic theories of mathematics both correctly. | 5 |
| PLO05 | Bilgi - Kuramsal, Olgusal | Understands the relationship between the different fields of mathematics and its relation to other disciplines. | 4 |
| PLO06 | Bilgi - Kuramsal, Olgusal | Comprehends the ability to understand the relationships between the objects in the most understandable way while creating a model for any problem. | |
| PLO07 | Bilgi - Kuramsal, Olgusal | Comprehend and explain mathematical models such as formulas, graphs, tables and schema. | 2 |
| PLO08 | Bilgi - Kuramsal, Olgusal | Demonstrate the ability to mathematically rearrange, analyze, and model the problems they encounter. | 5 |
| PLO09 | Bilgi - Kuramsal, Olgusal | Comprehends at least one of the computer programming languages. | |
| PLO10 | Bilgi - Kuramsal, Olgusal | Demonstrate the ability to use scientific methods and appropriate technologies effectively in problem solving. | 4 |
| PLO11 | Yetkinlikler - Bağımsız Çalışabilme ve Sorumluluk Alabilme Yetkinliği | Understands sufficient knowledge of foreign language to be able to understand Mathematical concepts and communicate with other mathematicians | |
| PLO12 | Yetkinlikler - Bağımsız Çalışabilme ve Sorumluluk Alabilme Yetkinliği | In addition to their professional development, they demonstrate their ability to continuously improve themselves by identifying their educational needs in scientific, cultural, artistic and social areas in line with their interests and abilities. | 2 |
| PLO13 | Yetkinlikler - Öğrenme Yetkinliği | Understands the programming techniques and shows the ability to do programming. | |
| PLO14 | Yetkinlikler - Öğrenme Yetkinliği | Demonstrates the ability to study mathematics both independently and as a group. | 5 |
| PLO15 | Bilgi - Kuramsal, Olgusal | Demonstrate an awareness of the universal and social impacts and legal consequences of mathematical applications in the field of study. | 4 |
| PLO16 | Bilgi - Kuramsal, Olgusal | Demonstrate the ability to select, use and develop effectively for contemporary mathematical applications. | 3 |
| PLO17 | Bilgi - Kuramsal, Olgusal | It has ability of lifelong learning awareness, access to information, monitoring developments in science and technology and self-renewal ability. | |
| PLO18 | Bilgi - Kuramsal, Olgusal | Gains the ability to use information technologies effectively for contemporary mathematical applications. | |
| PLO19 | Bilgi - Kuramsal, Olgusal | Gains the ability to design, conduct experiments, field work, data collection, analysis, archiving, text solving and / or interpretation according to mathematics fields. | 4 |
| PLO20 | Bilgi - Kuramsal, Olgusal | Gains the consciousness of prefesional ethics and responsibility. | 4 |
Week Plan
| Week | Topic | Preparation | Methods |
|---|---|---|---|
| 1 | Review of matric spaces. Definitions and examples. | Studying the relevant parts of the textbooks | Öğretim Yöntemleri: Anlatım |
| 2 | Relationship between convergence and continuity in metric spaces | Studying the relevant parts of the textbooks | Öğretim Yöntemleri: Anlatım |
| 3 | Problem solving 1 | Studying the relevant parts of the textbooks | Öğretim Yöntemleri: Alıştırma ve Uygulama, Problem Çözme |
| 4 | Cauchy sequences and completeness in metrik spaces. | Studying the relevant parts of the textbooks | Öğretim Yöntemleri: Anlatım |
| 5 | Some examples of complete and incomplete metric spaces. | Studying the relevant parts of the textbooks | Öğretim Yöntemleri: Anlatım |
| 6 | Banach Fixed Point Theorem | Studying the relevant parts of the textbooks | Öğretim Yöntemleri: Anlatım |
| 7 | Problem solving 2 | Studying the relevant parts of the textbooks | Öğretim Yöntemleri: Alıştırma ve Uygulama, Problem Çözme |
| 8 | Mid-Term Exam | Review and problem solving | Ölçme Yöntemleri: Yazılı Sınav |
| 9 | Review of basic properties of vector spaces. Definition of Norm. | Studying the relevant parts of the textbooks | Öğretim Yöntemleri: Anlatım |
| 10 | Normed spaces. Examples of normed spaces. | Studying the relevant parts of the textbooks | Öğretim Yöntemleri: Anlatım |
| 11 | Relationship between normed and metric spaces. Equivalent norms. | Studying the relevant parts of the textbooks | Öğretim Yöntemleri: Anlatım |
| 12 | Finite dimensional normed spaces. | Studying the relevant parts of the textbooks | Öğretim Yöntemleri: Anlatım |
| 13 | Convergence in normed spaces and norm of a linear transformation. | Studying the relevant parts of the textbooks | Öğretim Yöntemleri: Anlatım |
| 14 | Space of bounded Linear maps. Dual spaces | Studying the relevant parts of the textbooks | Öğretim Yöntemleri: Anlatım |
| 15 | Problem solving 3 | Review and problem solving | Öğretim Yöntemleri: Alıştırma ve Uygulama, Problem Çözme |
| 16 | Term Exams | Review and problem solving | Ölçme Yöntemleri: Yazılı Sınav |
| 17 | Term Exams | Review and problem solving | Ölçme Yöntemleri: Yazılı Sınav |
Student Workload - ECTS
| Works | Number | Time (Hour) | Workload (Hour) |
|---|---|---|---|
| Course Related Works | |||
| Class Time (Exam weeks are excluded) | 14 | 3 | 42 |
| Out of Class Study (Preliminary Work, Practice) | 14 | 3 | 42 |
| Assesment Related Works | |||
| Homeworks, Projects, Others | 0 | 0 | 0 |
| Mid-term Exams (Written, Oral, etc.) | 1 | 12 | 12 |
| Final Exam | 1 | 18 | 18 |
| Total Workload (Hour) | 114 | ||
| Total Workload / 25 (h) | 4,56 | ||
| ECTS | 5 ECTS | ||