Syllabus ( MATH 406 )

Basic information


Course title: 
Functional Analysis 
Course code: 
MATH 406 
Lecturer: 
Assoc. Prof. Dr. Ayşe SÖNMEZ

ECTS credits: 
7 
GTU credits: 
0 () 
Year, Semester: 
4, Spring 
Level of course: 
First Cycle (Undergraduate) 
Type of course: 
Compulsory

Language of instruction: 
English

Mode of delivery: 
Face to face

Pre and corequisites: 
Yok 
Professional practice: 
No 
Purpose of the course: 
This course aims to introduce metric spaces, topological spaces, normed and Banach spaces and teach the properties of these spaces, to introduce concepts related to operator theory, and to investigate the existence and uniqueness of operator equations. It also teaches some concepts such as dual spaces, HahnBanach extension theorem, weak convergence, inner product spaces and Hilbert spaces. 



Learning outcomes


Upon successful completion of this course, students will be able to:

Study the properties of metric and topological spaces. Explain and apply basic theorems.
Contribution to Program Outcomes

Having the knowledge about the scope, applications, history, problems, and methodology of mathematics that are useful to humanity both as a scientific and as an intellectual discipline.

Communicating between mathematics and other disciplines, and building mathematical models for interdisciplinary problems.

Having improved abilities in mathematics communications, problemsolving, and brainstorming skills.

Exhibiting professional and ethical responsibility.
Method of assessment

Written exam

Study a properties of Euclid, Hilbert and Banach spaces. Explain and apply basic theorems
Contribution to Program Outcomes

Having the knowledge about the scope, applications, history, problems, and methodology of mathematics that are useful to humanity both as a scientific and as an intellectual discipline.

Having improved abilities in mathematics communications, problemsolving, and brainstorming skills.
Method of assessment

Written exam

Define Dual spaces and analyze properties of linear functions and opeators. Explain and apply basic theorems.
Contribution to Program Outcomes

Having the knowledge about the scope, applications, history, problems, and methodology of mathematics that are useful to humanity both as a scientific and as an intellectual discipline.

Having improved abilities in mathematics communications, problemsolving, and brainstorming skills.
Method of assessment

Written exam


Contents


Week 1: 
Complete metric spaces, closure. 
Week 2: 
Contraction mapping. 
Week 3: 
Metric and Topological spaces. 
Week 4: 
Compactness. Arzely theorem. 
Week 5: 
Linear normed and topological spaces. 
Week 6: 
Convex functional and Minkowski functional. 
Week 7: 
Euclidean spaces. 
Week 8: 
Midterm exam and solutions. 
Week 9: 
Fourier Series.RieszFischer theorem. 
Week 10: 
Inner product spaces. 
Week 11: 
Dual spaces. The Riesz representation theorem. 
Week 12: 
Strong and Weak Topologies. Strong and Weak Convergence. 
Week 13: 
Bounded and Compact operators. 
Week 14: 
Spectrum and resolvent 
Week 15*: 
 
Week 16*: 
Final exam. 
Textbooks and materials: 
• A.N.Kolmogorov, S.V. Fomin,Elements of the Theory of Functions and Functional Analysis,Dover Pubns, 1999. • E. Kreyzig, Introductory Functional Analysis with Applications, John Wiley&Sons, 1978. • L.A.Lusternik, V.J.Sobolev, “Elements of Functional Analysis”. • Walter Rudin, “Functional Analysis”. • John B. Conway “A Course in Functional Analysis”. • R.E. Edwards, “Functional Analysis : Theory and Applications”.

Recommended readings: 
• A.N.Kolmogorov, S.V. Fomin,Elements of the Theory of Functions and Functional Analysis,Dover Pubns, 1999. • E. Kreyzig, Introductory Functional Analysis with Applications, John Wiley&Sons, 1978. • L.A.Lusternik, V.J.Sobolev, “Elements of Functional Analysis”. • Walter Rudin, “Functional Analysis”. • John B. Conway “A Course in Functional Analysis”. • R.E. Edwards, “Functional Analysis : Theory and Applications”.


* Between 15th and 16th weeks is there a free week for students to prepare for final exam.




Assessment



Method of assessment 
Week number 
Weight (%) 

Midterms: 
8 
40 
Other interm studies: 

0 
Project: 

0 
Homework: 

0 
Quiz: 

0 
Final exam: 
16 
60 

Total weight: 
(%) 



Workload



Activity 
Duration (Hours per week) 
Total number of weeks 
Total hours in term 

Courses (Facetoface teaching): 
3 
14 

Own studies outside class: 
7 
14 

Practice, Recitation: 
0 
0 

Homework: 
0 
0 

Term project: 
0 
0 

Term project presentation: 
0 
0 

Quiz: 
0 
0 

Own study for midterm exam: 
12 
1 

Midterm: 
3 
1 

Personal studies for final exam: 
18 
1 

Final exam: 
2 
1 



Total workload: 



Total ECTS credits: 
* 

* ECTS credit is calculated by dividing total workload by 25. (1 ECTS = 25 work hours)



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