Iğdır University Lifelong Learning Programme

Description of Individual Course Units

Course Unit Code	Course Unit Title	Type of Course Unit	Year of Study	Semester	Number of ECTS Credits
MAT-23-206	FunctIonal AnalysIs II	Elective	1	2	6

Level of Course Unit

Second Cycle

Objectives of the Course

The purpose of this course is to explain the dual spaces of the Banach and Hilbert spaces, their properties, and the basic concepts of the linear transformations (operators, functionals) theory defined in these spaces, and their applications. For example, Lebesgue, Sobolev et al. space and differential, integral and other linear transformations.The students can understand the works about these themes.

Name of Lecturer(s)

Prof. Dr. Kamal SOLTANOV

Learning Outcomes

1	To explain to students why Banach and Hilbert spaces, their properties and their properties are important
2	To teach the students the space of functionals, dual spaces and their properties: in the abstract case and examples
3	To teach students Lebesgue and Sobolev spaces; linear bounded and unbounded operators (and functionals) defined on these spaces, and to explain of their properties
4	To explain the basic results of the theory of Banach spaces and of theory of operators defined on these spaces (general theorems) in general cases and examples.

Mode of Delivery

Daytime Class

Prerequisites and co-requisities

It is need to know the basic facts of Mathematical Analysis, Linear Algebra, Real analysis, Ordinary and Partial Differential Equations, Algebra and Topology in the graduate level.

Recommended Optional Programme Components

Course Contents

Orthogonal system, orthogonalization, orthonormality in Hilbert spaces. Linear functionals and Hyperplanes Application of the corollaries of Hahn-Banach theorem to the Lebesgue spaces Application of the corollaries of Hahn-Banach theorem to the Sobolev spaces. Examples. The geometrik form of the Hahn-Banach theorem and corollaries, Examples. Banach-Steinhaus theorem Banach fixed-point theorem and its applications Spaces of the bounded and unbounded operators defined in the Banach spaces. Examples. The weak topology in the Banach spaces and its properties The weak topology and weak compactness in the Banach spaces Reflexive spaces and their properties Operator equations in the Banach space

Weekly Detailed Course Contents

Week	Theoretical	Practice	Laboratory
1	Orthogonal system, orthogonalization, orthonormality in Hilbert spaces.
2	Linear functionals and Hyperplanes
3	Application of the corollaries of Hahn-Banach theorem to the Lebesgue spaces
4	Application of the corollaries of Hahn-Banach theorem to the Sobolev spaces. Examples.
5	The geometrik form of the Hahn-Banach theorem and corollaries, Examples.
6	Corollaries of the Hahn-Banach theorem. Examples
7	MID TERM EXAM
8	Banach-Steinhaus theorem
9	Banach fixed-point theorem and its applications
10	Spaces of the bounded and unbounded operators defined in the Banach spaces. Examples.
11	The weak topology in the Banach spaces and its properties
12	The weak topology and weak compactness in the Banach spaces
13	Reflexive spaces and their properties
14	Operator equations in the Banach space and preparation to the final exam
15	Final Exam

Recommended or Required Reading

Lusternik, L. A; Sobolev, V. J: Elements of Functional Analysis. Wiley 1974. Yoshida, K: Functional Analysis. Springer Verlag 1980. Rudin, W : Functional Analysis. Mc Graw Hill 1985. Kirillov A., Gvishiani A. D.: Theorems and Problems in Functional Analysis. Springer-Verlag, New York, 1982 Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations, 2011, Springer, N.Y.

Planned Learning Activities and Teaching Methods

Assessment Methods and Criteria

Term (or Year) Learning Activities	Quantity	Weight
Midterm Examination	1	100
SUM		100
End Of Term (or Year) Learning Activities	Quantity	Weight
Final Examination	1	100
SUM		100
Term (or Year) Learning Activities		50
End Of Term (or Year) Learning Activities		50
SUM		100

Language of Instruction

Turkish

Work Placement(s)

Workload Calculation

Activities	Number	Time (hours)	Total Work Load (hours)
Midterm Examination	1	1	1
Final Examination	1	2	2
Criticising Paper	14	3	42
Self Study	14	3	42
Individual Study for Mid term Examination	1	50	50
Individual Study for Final Examination	1	50	50
TOTAL WORKLOAD (hours)			187

Contribution of Learning Outcomes to Programme Outcomes

	PO 1	PO 2	PO 3	PO 4	PO 5	PO 6	PO 7	PO 8	PO 9	PO 10	PO 11
LO1	5	5	5	5	5	5	5	5	5	5	5
LO2	5	5	5	5	5	5	5	5	5	5	5
LO3	5	5	5	5	5	5	5	5	5	5	5
LO4	5	5	5	5	5	5	5	5	5	5	5

* Contribution Level : 1 Very low 2 Low 3 Medium 4 High 5 Very High