# Functional Analysis (AN4)

### General

School School of Science Department of Mathematics Graduate AN4 1 Functional Analysis Lectures (Weekly Teaching Hours: 3, Credits: 7.5) General Background - Greek Yes (in English) 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. 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
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.