Functional Analysis (AN4)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	AN4
Semester	1
Course Title	Functional Analysis
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	General Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	The objectives of the course are: The acquisition of background from the students on the basic structures and techniques of Functional Analysis, as independent knowledge as well as a tool for the other branches of Analysis, so that they will have the potential to apply the knowledge they get in applications.
General Competences	The course aims to give the postgraduate student the ability to analyse and synthesize advanced concepts of Functional Analysis. The goal is to acquire the skills for autonomous work and teamwork in an interdisciplinary environment and to able to produce new research ideas.

Syllabus

1. Normes spaces, Banach spaces and Hilbert spaces, classical examples (sequence spaces and function spaces). Basic theorems.
2. General theory of topological vector spaces, locally convex spaces, separation theorems.
3. Weak topologies, theorems of Mazur, Alaoglu and Goldstine, weak compactness.
4. Schauder bases and basic sequences.
5. Extreme points, Krein Milman theorem.
6. Riesz representation theorem, Lp spaces.
7. Fixed point theorems.

Teaching and Learning Methods - Evaluation

Delivery	Face to face
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Study and analysis of bibliography 78 Preparation of assignments and interactive teaching 70.5 Course total 187.5
Student Performance Evaluation	Written examination at the end of the semester.

Attached Bibliography

• J. Bak and D. J. Newman, Complex Analysis (3rd ed.), Springer, 2010.
• S. Lang, Complex Analysis (4th ed.), Springer, 1999.
• I. Markushevich, Theory of Functions of a Complex Variable (2nd ed.), Vol. 1-3, AMS Chelsea, 2011.
• I. Markushevich, The Theory of Analytic Functions: A Brief Course, Mir Publishers, 1983.
• R. Remmert, Theory of Complex Functions, Springer, 1990.
• R. Remmert, Classical Topics in Complex Function Theory, Springer, 1998.
• K. Jänich, Funktionentheorie (6te Aufl.), Springer, 2011.