PAÜ .:. Eğitim Öğretim Bilgi Sistemi

COURSE INFORMATION

Course Code	Course Title	L+P Hour	Semester	ECTS
MAT 540	FUNCTIONAL ANALYSIS I	3 + 0	2nd Semester	7,5

COURSE DESCRIPTION

Course Level	Master's Degree
Course Type	Elective
Course Objective	The aim of this course is to teach concepts of metric space, norm space, Banach space, Hilbert space and basic theorems on Banach Space.
Course Content	Metric Spaces, Normed Spaces, Banach Spaces, Inner Product Spaces, Hilbert Spaces, Fundamental Theorems on Normed and Banach Spaces, Advanced Application, Banach Fixed Point Theorem, Approach Theory.
Prerequisites	No the prerequisite of lesson.
Corequisite	No the corequisite of lesson.

COURSE LEARNING OUTCOMES

1	Learns the structures of Metric Spaces, Normed Spaces, Banach Spaces, Inner Product Spaces, Hilbert Spaces.
2	Knows the expression and proof of the Fundamental Theorems on Normed and Banach Spaces.
3	States and proves the Banach fixed point theorem.
4	Have information on Approximation Theory.

COURSE'S CONTRIBUTION TO PROGRAM

ECTS ALLOCATED BASED ON STUDENT WORKLOAD BY THE COURSE DESCRIPTION

Activities	Quantity	Duration (Hour)	Total Work Load (Hour)
Course Duration (14 weeks/theoric+practical)	14	3	42
Hours for off-the-classroom study (Pre-study, practice)	14	7	98
Assignments	1	5	5
Mid-terms	1	15	15
Final examination	1	35	35
Total Work Load ECTS Credit of the Course			195 7,5

COURSE DETAILS

Select Year

Course Details

Course Code	Course Title	L+P Hour	Course Code	Language Of Instruction	Course Semester	Mode of Delivery
MAT 540	FUNCTIONAL ANALYSIS I	3 + 0	1	Turkish	2025-2026 Spring	Face to Face

Course Coordinator	E-Mail	Phone Number	Course Location	Attendance
Prof. Dr. ÖZLEM GİRGİN ATLIHAN	oatlihan@pau.edu.tr		FEN A0216	%90

Goals		The aim of this course is to teach concepts of metric space, norm space, Banach space, Hilbert space and basic theorems on Banach Space.
Content		Metric Spaces, Normed Spaces, Banach Spaces, Inner Product Spaces, Hilbert Spaces, Fundamental Theorems on Normed and Banach Spaces, Advanced Application, Banach Fixed Point Theorem, Approach Theory.

Topics

Weeks	Topics
1	Linear Operator, Kernel of a Linear Operator, Isomorphism
2	Bounded and Continuous Linear Operators
3	Bounded and Continuous Linear Operators
4	Bounded Linear Extensions, Linear Functionals and Dual Space
5	Algebraic Dual Space, Linear Operators and Functionals on Finite Dimensional Spaces
6	Hahn-Banach Theorem, Hahn-Banach Theorem for Complex Linear Spaces and Normed Spaces, Baire Theorem
7	Open Mapping Theorem, Closed Linear Operators and Closed Graph Theorem
8	Banach-Steinhouse Theorem, Adjoint Operator for Normed Spaces
9	Midterm Exam
10	Inner Product Space, Parallelogram Equality, Orthogonality, Pythagorean Theorem
11	Hilbert Space, Orthogonal Complement and Direct Sum
12	Closed Subspace, Minimizing Vector Theorem, Orthogonal Projection Operator
13	Riesz’s Theorem, Hilbert-adjoint Operator
14	Self-Adjoint and Normal Operators

Materials

Materials are not specified.

Resources

Resources	Resources Language
FONKSİYONEL ANALİZ,Prof.Dr.Mustafa BAYRAKTAR	Türkçe

Course Assessment

Assesment Methods	Percentage (%)	Assesment Methods Title
Final Exam	50	Final Exam
Midterm Exam	50	Midterm Exam

L+P: Lecture and Practice
PQ: Program Learning Outcomes
LO: Course Learning Outcomes

Prospective Student