# Functional Analysis I (MAE851)

### General

School School of Science Department of Mathematics Undergraduate MAE851 8 Functional Analysis I Lectures (Weekly Teaching Hours: 3, Credits: 6) Special Background - Greek Yes ( in English) See eCourse, the Learning Management System maintained by the University of Ioannina.

### Learning Outcomes

Learning outcomes The goal of this course is: To familiarize the student with the notions the basic theorems and the techniques concerning Banach spaces, bounded linear operators between them, dual spaces and especially Hilbert spaces. After completing this course the student will be able to recognize if a given normed linear space is a Banach space, to compute the norm of a bounded linear operator, will be able to use the basic theorems of Functional analysis (Hahn-Banach theorem and its consequences, Open mapping theorem, Uniform Boundedness Principle), and will get the basic theorems and techniques concerning Hilbert spaces (e.g. existence of orthonormal bases, Gram-Schmidt orthogonalization procedure, isometry of every Hilbert space with its dual). This course aims to provide the student with the theoretical background and the fluency of using the basic theorems and techniques of Functional Analysis. Promotes the analytical and synthetic thinking that the student will be able to apply the knowledge acquired in a broader scope including the whole range of mathematical analysis.

### Syllabus

Linear spaces and algebraic bases (Hamel bases), linear operators. Normed spaces, Banach spaces, classical examples. Bounded linear operators, dual spaces, conjugate operators. Hahn Banach theorem and its consequences. Reflexive spaces. Inner product spaces, Hilbert spaces, orthonormal systems, every Hilbert space is isometric with its dual. Baire's category theorem and some of its consequences in Functional Analysis (Open Mapping Theorem, Closed graph Theorem, Uniform Boundedness Principle, Banach Steinhauss Theorem).

### Teaching and Learning Methods - Evaluation

Delivery

Teaching on the blackboard from the teacher

Use of Information and Communications Technology -
Teaching Methods