MTK709 - LINEAR EQUATIONS IN BANACH SPACES

Course Name Code Semester Theory
(hours/week)		 Application
(hours/week)		 Credit ECTS
LINEAR EQUATIONS IN BANACH SPACES MTK709 1st Semester 3 0 3 12
PrequisitesMTK242 Linear Algebra II, MTK111 Calculus for Math Students I, MTK112 Calculus for Math Students II, MTK201 Advanced Calculus, MTK141 Abstract Mathematics I, MTK413 Fundamentals of the Theory of Functions and Functional Analysis, MTK203 Differential Equations,MTK302 Partial Differential Equations, M
Course languageTurkish
Course typeElective 
Mode of DeliveryFace-to-Face 
Learning and teaching strategiesLecture
Discussion
Question and Answer
Problem Solving
 
Instructor (s)Instructors of the department of mathematics Prof.Dr. Kamal Soltanov  
Course objectiveThe aim of this course is to teach to students, Banach spaces and its duals, linear equations in Banach spaces and how and in what sense it can be solved, explanation of Adjoint Operators in Banach spaces and Adjoint Equations; Relation between solvability of given Equation and its dual, Fredholm and Noether Equations and Applications of the given general results. 
Learning outcomes
  1. Understands Banach spaces, its duals, properties and relations;
  2. Explains linear operators , its duals and properties (as closedness, closable, kernel) in Banach spaces;
  3. Knows in what conditions Linear equation can be solvable everywhere, densely, normally and uniquely in Banach spaces;
  4. Explains relations between solvability of Linear Equation and kernel of dual;
  5. Identifies relations between Factored Equation and given Equation;
  6. Explains Fredholm alternative
Course ContentBanach spaces, its duals, properties and relations;
Linear Operators and its duals in Banach spaces, explanation of its properties (as closedness, closable, kernel);
The sufficiently and necessary conditions in what Linear Equation can be solvable everywhere, densely, normally and uniquely in Banach spaces;
Relations between solvability of Linear Equation and its dual Equation;
Defect of Equation, index of Equation (Operator);
Fredholm Alternative
 
ReferencesDunford N., Schwartz J. Linear Operators. I:General Theory, Interscince, N-Y-L, 1967
Krein S. G. Linear Equations in Banach Spaces. Birkhauser, 1982
Liusternik L. A., Sobolev V. I. Elements of Functional Analysis. 1961
Yosida K. - Functional Analysis. Springer-Verlag, 1980
Lions J.-L. - Magenes E. Non-homogeneus boundary value problems and Applications. Springer Verlag, 1972.
Brezis H. - Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, 2011
 

Course outline weekly

WeeksTopics
Week 1Banach spaces, its duals, properties and topological relations. Examples
Week 2Linear Operators and equations in Banach spaces, Examples
Week 3Equations with a Closed Operators, Examples, Assignments
Week 4The Adjoint Equation, Factored Equation and Adjoint, Examples
Week 5Equation with a Closed Operator which has a dense domain, Examples, Assignments
Week 6Midterm exam
Week 7Normally solvable Equations with finite dimensional kernel, Examples,
Week 8Equations with finite Defect, Examples, Assignments
Week 9n-Normal and d-normal Equations, Examples
Week 10Noetherian Equations, Index, Examples
Week 11Midterm exam
Week 12Equations with Operators which act in a single space, Examples
Week 13Fredholm Equations, Regularization of Equations, Examples, Assignments
Week 14Differential and Integral Equations, Examples
Week 15Prep. final exam
Week 16Final Exam

Assesment methods

Course activitiesNumberPercentage
Attendance00
Laboratory00
Application00
Field activities00
Specific practical training00
Assignments830
Presentation00
Project00
Seminar00
Midterms220
Final exam150
Total100
Percentage of semester activities contributing grade succes050
Percentage of final exam contributing grade succes050
Total100

WORKLOAD AND ECTS CALCULATION

Activities Number Duration (hour) Total Work Load
Course Duration (x14) 14 3 42
Laboratory 0 0 0
Application000
Specific practical training000
Field activities000
Study Hours Out of Class (Preliminary work, reinforcement, ect)1415210
Presentation / Seminar Preparation000
Project000
Homework assignment8972
Midterms (Study duration)2816
Final Exam (Study duration) 12020
Total Workload3955360

Matrix Of The Course Learning Outcomes Versus Program Outcomes

D.9. Key Learning OutcomesContrubition level*
12345
1. Deepens the concepts of mathematics in the level of expertise.  X  
2. Grasps the inter-disciplinary interaction related to the area; reaches original results by using the specialist knowledge in analyzing and evaluating new ideas.  X  
3. Gains the ability to think independently and develops theoretical concepts.      
4. Develops original mathematical models by using interrelations between mathematics and other disciplines and applies them to other disciplines.  X  
5. Uses high level research methods in studies in the area.  X  
6. Develops a new idea, method and/or application independently, finds a solution, and contributes to the progress in the area by carrying out original studies.     
7. Fulfills the leader role in the environments where solutions are thought for the area and/or inter-disciplinary problems.  X  
8. Develops continually the skills of creativity, decision making and problem solving.  X  
9. Defends original opinions by communicating with experts in the area.     
10. Uses a foreign language- at least C1 Level-, communicates with foreign colleagues and follows the international literature.     
11. Follows the latest developments in the information and communication technologies and uses them in the area.   X 
12. Does research in national and international research groups.  X  
13. Makes strategic decision in the solution of problems in the area.  X  
14. Protects the rights of other researchers in regards to ethics, privacy, ownership and copyright.   X 

*1 Lowest, 2 Low, 3 Average, 4 High, 5 Highest