COURSE UNIT TITLE

: INTRODUCTION TO FUNCTIONAL ANALYSIS

DEGREE PROGRAMMES

Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS
MAT 4059 INTRODUCTION TO FUNCTIONAL ANALYSIS ELECTIVE 4 0 0 7

Offered By

Mathematics

Level of Course Unit

First Cycle Programmes (Bachelor's Degree)

Course Coordinator

ASSOCIATE PROFESSOR SEDEF KARAKILIÇ

Offered to

Mathematics (Evening)
Mathematics

Course Objective

It aims to familiarize the basic concepts and principles of functional analysis such as Banach and Hilbert spaces, linear bounded operators and functionals, Riesz Theorem, Banach Theorem, Open Mapping Theorem, Uniform Boundedness Theorem, compact operators and their spectrum.

Learning Outcomes of the Course Unit

1   to relate Hilbert spaces, Banach spaces and metric spaces.
2   to distinguish an orthonormal basis for a Hilbert space.
3   to relate l_2 and the other Hilbert spaces.
4   to understand Riesz Theorem.
5   to understand the fundamental theorems of functional analysis: Banach Theorem, Open Mapping Theorem, Uniform Boundedness Theorem.
6   to use the spectral properties of a self adjoint compact operator.
7   to distinguish between weak convergence, strong convergence and uniform convergence.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description
1 Review of metric spaces. The sequence spaces l_p.
2 Banach spaces. Examples of Banach spaces. Subspaces of a normed linear space.
3 Inner product spaces. Examples of Hilbert spaces. Cauchy-Schwarz inequality. Pythagorean relation.
4 Orthogonality and bases. Bessel's inequality. Parseval's equality.
5 Orthogonal complements and direct sum. The Isomorphism Theorem.
6 Continuous functionals. Riesz Theorem. Definitions and examples of linear operators. Continuity and boundedness of linear operators.
7 Invertibility of an operator. Banach's Theorem.
8 Spectrum of a bounded linear operator.
9 The adjoint of an operator. The adjoint of an operator in a Hilbert space. Self adjoint operators.
10 Weak and strong convergence. Uniform convergence.
11 Principle of uniform boundedness. Open Mapping Theorem.
12 Compact operators on Hilbert spaces.
13 Eigenvalues of compact self adjoint operators.
14 Spectral representations of compact self adjoint operators.

Recomended or Required Reading

Textbook(s): Linear Functional Analysis (second edition), Bryan P. Rynne, Martin A. Youngson, Springer-Verlag, 2008.
Supplementary Book(s): Introductory to Functional Analysis with Applications, Erwin Kreyszig, John Wiley & Sons, 1978.
References: Introductory Real Analysis, A. N. Kolmogorov, S.V. Fomin, Dover Publications, 1970.

Planned Learning Activities and Teaching Methods

Lecture Notes
Text Book(s)
Solving Problems

Assessment Methods

SORTING NUMBER SHORT CODE LONG CODE FORMULA
1 MTE 1 MIDTERM EXAM 1
2 MTE 2 MIDTERM EXAM 2
3 FIN FINAL EXAM
4 FCG FINAL COURSE GRADE MTE1 * 0.30 + MTE2 * 0.30 + FIN * 0.40
5 RST RESIT
6 FCGR FINAL COURSE GRADE (RESIT) MTE1 * 0.30 + MTE2 * 0.30 + RST * 0.40


*** Resit Exam is Not Administered in Institutions Where Resit is not Applicable.

Further Notes About Assessment Methods

None

Assessment Criteria

%30(mt1)+%30(mt2)+%40(final)

Language of Instruction

English

Course Policies and Rules

You can be successful in this course by studying from your textbooks and lecture
notes on the topics to be covered every week, coming to class by solving the given
problems, establishing the concepts by discussing the parts you do not understand
with your questions, learning the methods, and actively participating in the course.

Contact Details for the Lecturer(s)

e-mail: sedef.erim@deu.edu.tr tel: (232) 301 85 91

Office Hours

to be announced later

Work Placement(s)

None

Workload Calculation

Activities Number Time (hours) Total Work Load (hours)
Lectures 14 4 56
Preparations before/after weekly lectures 14 4 56
Preparation for midterm exam 2 12 24
Preparation for final exam 1 26 26
Final 1 2 2
Midterm 2 2 4
TOTAL WORKLOAD (hours) 168

Contribution of Learning Outcomes to Programme Outcomes

PO/LOPO.1PO.2PO.3PO.4PO.5PO.6PO.7PO.8PO.9PO.10PO.11PO.12PO.13
LO.15544534
LO.25544534
LO.35544534
LO.45544534
LO.55544534
LO.65544534
LO.75544534

CONTACT INFORMATION
Dokuz Eylül Üniversitesi Cumhuriyet Bulvarı No: 144 35210 Alsancak / İZMİR
Phone: +90(232) 412 12 12 - Fax: +90 (232) 464 81 35

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