MATH 442 | Course Introduction and Application Information - Department of Mathematics

Course Name

Functional Analysis II

Code	Semester	Theory (hour/week)	Application/Lab (hour/week)	Local Credits	ECTS
MATH 442	Fall/Spring	3	0	3	7

Prerequisites

MATH 441 To attend the classes (To enrol for the course and get a grade other than NA or W)

Course Language

English

Course Type

Elective

Course Level

First Cycle

Mode of Delivery

face to face

Teaching Methods and Techniques of the Course

Problem Solving
Q&A
Lecture / Presentation

National Occupation Classification

-

Course Coordinator

-

Course Lecturer(s)

Öğr. Gör. Dr. Tuğçe KATICAN

Assistant(s)

Course Objectives

This course provides a deep understanding of introductory functional analysis. The objective of this course is to cover fundamental theorems of functional analysis such as Hahn-Banach theorem, Open mapping theorem, Closed graph theorem, Baire’s category theorem, Banach fixed point theorem, and their applications.

Learning Outcomes

The students who succeeded in this course;

explain the relationship between inner product spaces, Hilbert space, Banach space and metric space.
solve problems using properties of orthogonal complement and direct sum.
discuss the orthogonality of sequences and sets, and their properties.
solve problems by using properties of Hilbert adjoint operator, self- adjoint operator, identity and normal operators.
apply the basic theorems of functional analysis.
compare the concepts of strong and weak convergence.
explain the differences between the convergence of sequences, functions and operators.

Course Description

This course aims to teach basic theory and applications of Functional Analysis.

Course Category	Core Courses	X
	Major Area Courses
	Supportive Courses
	Media and Management Skills Courses
	Transferable Skill Courses

Week	Subjects	Related Preparation	Learning Outcome
1	Normed and Banach spaces (Review)	Erwin Kreyszig,“Introductory Functional Analysis with Applications”, (Wiley,1989). Section: 2
2	Inner product spaces and their properties, Hilbert spaces	Erwin Kreyszig,“Introductory Functional Analysis with Applications”, (Wiley,1989). Section: 3.2, 3.3
3	Orthogonal complements and direct sums	Erwin Kreyszig,“Introductory Functional Analysis with Applications”, (Wiley,1989). Section: 3.3
4	Orthogonal complements and direct sums	Erwin Kreyszig,“Introductory Functional Analysis with Applications”, (Wiley,1989). Section: 3.4
5	Orthonormal sets and sequences	Erwin Kreyszig,“Introductory Functional Analysis with Applications”, (Wiley,1989). Section: 3.8
6	Fourier series and their properties	Erwin Kreyszig,“Introductory Functional Analysis with Applications”, (Wiley,1989). Section: 3.9
7	Total orthonormal sets and sequences	Erwin Kreyszig,“Introductory Functional Analysis with Applications”, (Wiley,1989). Section: 3.10
8	Representation of functionals on Hilbert spaces	Erwin Kreyszig,“Introductory Functional Analysis with Applications”, (Wiley,1989). Section: 4.1, 4.2
9	Hilbert-Adjoint operator	Erwin Kreyszig,“Introductory Functional Analysis with Applications”, Wiley,1989. Section: 4.2, 4.3
10	Self-adjoint, unitary and normal operators	Erwin Kreyszig,“Introductory Functional Analysis with Applications”, (Wiley,1989). Section: 4.7
11	Fundamental theorems of functional analysis: Zorn's lemma, Hahn-Banach theorem and Banach fixed point theorem	Erwin Kreyszig,“Introductory Functional Analysis with Applications”, (Wiley,1989). Section: 4.7,
12	Fundamental theorems of functional analysis: Baire category theorem, uniform boundedness theorem, open mapping theorem and Banach fixed point theorem	Erwin Kreyszig,“Introductory Functional Analysis with Applications”, (Wiley,1989). Section: 4.13
13	Weak and strong convergence	Erwin Kreyszig,“Introductory Functional Analysis with Applications”, (Wiley,1989). Section: 4.8
14	Convergence of sequences of operators and functionals	Erwin Kreyszig,“Introductory Functional Analysis with Applications”, (Wiley,1989). Section: 4.9
15	Semester review
16	Final exam

Course Notes/Textbooks

Erwin Kreyszig, “Introductory Functional Analysis with Applications” , (Wiley, 1989).

ISBN-13: 978-0471504597

Suggested Readings/Materials

Walter Rudin ,'' Functional Analysis'', 2nd edition

(McGraw-Hill Science/Engineering/Math,1991).

ISBN-13:978-0070542365

Semester Activities	Number	Weigthing
Participation
Laboratory / Application
Field Work
Quizzes / Studio Critiques
Portfolio
Homework / Assignments	1	20
Presentation / Jury
Project
Seminar / Workshop	1	20
Oral Exams
Midterm
Final Exam	1	60
Total

Weighting of Semester Activities on the Final Grade	2	40
Weighting of End-of-Semester Activities on the Final Grade	1	60
Total

Semester Activities	Number	Duration (Hours)	Workload
Theoretical Course Hours (Including exam week: 16 x total hours)	16	3	48
Laboratory / Application Hours (Including exam week: '.16.' x total hours)	16		0
Study Hours Out of Class	14	4	56
Field Work			0
Quizzes / Studio Critiques			0
Portfolio			0
Homework / Assignments	1	36	36
Presentation / Jury			0
Project			0
Seminar / Workshop			0
Oral Exam			0
Midterms			0
Final Exam	1	70	70
		Total	210

#	PC Sub	Program Competencies/Outcomes	* Contribution Level
#	PC Sub	Program Competencies/Outcomes	1	2	3	4	5
1	To be able to have a grasp of basic mathematics, applied mathematics or theories and applications of statistics.		-	-	-	X	-
2	To be able to use advanced theoretical and applied knowledge, interpret and evaluate data, define and analyze problems, develop solutions based on research and proofs by using acquired advanced knowledge and skills within the fields of mathematics or statistics.		-	-	-	-	-
3	To be able to apply mathematics or statistics in real life phenomena with interdisciplinary approach and discover their potentials.		-	-	-	-	X
4	To be able to evaluate the knowledge and skills acquired at an advanced level in the field with a critical approach and develop positive attitude towards lifelong learning.		-	-	-	-	-
5	To be able to share the ideas and solution proposals to problems on issues in the field with professionals, non-professionals.		-	-	-	-	-
6	To be able to take responsibility both as a team member or individual in order to solve unexpected complex problems faced within the implementations in the field, planning and managing activities towards the development of subordinates in the framework of a project.		-	-	X	-	-
7	To be able to use informatics and communication technologies with at least a minimum level of European Computer Driving License Advanced Level software knowledge.		-	-	-	-	X
8	To be able to act in accordance with social, scientific, cultural and ethical values on the stages of gathering, implementation and release of the results of data related to the field.		-	-	-	-	-
9	To be able to possess sufficient consciousness about the issues of universality of social rights, social justice, quality, cultural values and also environmental protection, worker's health and security.		-	-	-	-	-
10	To be able to connect concrete events and transfer solutions, collect data, analyze and interpret results using scientific methods and having a way of abstract thinking.		-	-	-	-	-
11	To be able to collect data in the areas of Mathematics or Statistics and communicate with colleagues in a foreign language.		-	-	-	-	-
12	To be able to speak a second foreign language at a medium level of fluency efficiently.		-	-	-	-	-
13	To be able to relate the knowledge accumulated throughout the human history to their field of expertise.		-	-	-	-	-