FUNCTIONAL ANALYSIS

Course Code:

MATH 456

Semester:

Spring

Course Type:

Core

P:

2

Lab:

2

Laboratuvar Saati:

0

Credits:

3

ECTS:

9

Prerequisite Courses:

METRIC AND TOPOLOGICAL SPACES

Course Language:

English

Course Objectives:

Functional analysis, is a subject that has many applications. We can count the theory of differential equations and applications in physics among them.

Course Content:

Topological dual. Compact, closed and adjoint operators. Inner product spaces. Orthonormal sets and Fourier series. Linear operators on Hilbert spaces. Resolvent and spectrum of an operator. Spectra of continuous and compact linear operators. Spectral analysis on Hilbert spaces. Derivations of operators.

Course Methodology:

1: Lecture, 2: Problem Solving

Course Evaluation Methods:

A: Written examination, B: Homework

Vertical Tabs

Course Learning Outcomes

Learning Outcomes	Teaching Methods	Assessment Methods
1) Learns inner product and Hilbert Spaces	1,2	A
2) Computes the Fourier Coefficients with respect to an orthonormal basis	1,2	A
3) Learns dual spaces and to utilize of Hahn-Banach theorem	1,2	A
4) Learns Riesz Representation Theorem	1,2	A
5) Learns the spectrum of linear operators	1,2	A
6) Learns compact operators and how to apply them	1,2	A

Course Flow

Week	Topics	Study Materials
1	Inner Product Spaces, Hilbert Spaces ,Orthogonality
2	Orthonormal Bases in Inﬁnite Dimensions ,Fourier Series
3	Continuous Linear Transformations
4	Hahn–Banach Theorem
5	Dual Spaces
6	The Second Dual, Reﬂexive Spaces and Dual Operators
7	Projections and Complementary Subspaces
8	Linear Operators on Hilbert Spaces, Riesz Theorem
9	The Adjoint of an Operator
10	Normal, Self-adjoint and Unitary Operators
11	The Spectrum of an Operator
12	Positive Operators and Projections
13	Compact Operators
14	Spectral Theory of Compact Operators

Recommended Sources

Textbook	Linear Functional Analysis, Bryan Rynne, M.A. Youngson
Additional Resources

Material Sharing

Documents
Assignments
Exams

Assessment

IN-TERM STUDIES	NUMBER	PERCENTAGE
Mid-terms	2	100
Quizzes	0	0
Assignments	0	0
Total		100
CONTRIBUTION OF FINAL EXAMINATION TO OVERALL GRADE		60
CONTRIBUTION OF IN-TERM STUDIES TO OVERALL GRADE		40
Total		100

COURSE CATEGORY

Expertise/Field Courses

Course’s Contribution to Program

No	Program Learning Outcomes	Contribution
No	Program Learning Outcomes	1	2	3	4	5
1	The ability to make computation on the basic topics of mathematics such as limit, derivative, integral, logic, linear algebra and discrete mathematics which provide a basis for the fundamenral research fields in mathematics (i.e., analysis, algebra, differential equations and geometry)					X
2	Acquiring fundamental knowledge on fundamental research fields in mathematics					X
3	Ability form and interpret the relations between research topics in mathematics					X
4	Ability to define, formulate and solve mathematical problems					X
5	Consciousness of professional ethics and responsibilty			X
6	Ability to communicate actively			X
7	Ability of self-development in fields of interest					X
8	Ability to learn, choose and use necessary information technologies	X
9	Lifelong education			X

ECTS

ECTS ALLOCATED BASED ON STUDENT WORKLOAD BY THE COURSE DESCRIPTION
Activities	Quantity	Duration (Hour)	Total Workload (Hour)
Course Duration (14x Total course hours)	14	4	56
Hours for off-the-classroom study (Pre-study, practice)	14	7	98
Mid-terms (Including self study)	2	23	46
Quizzes	-
Assignments	-
Final examination (Including self study)	1	25	25
Total Work Load			225
Total Work Load / 25 (h)			9
ECTS Credit of the Course			9

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