Functional Analysis (MAE719)

Undergraduate Courses Outlines - Department of Mathematics

General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE711
Semester	7
Course Title	Functional Analysis I
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 6)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes ( in English)
Course Website (URL)	-

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).

General Competences

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	Activity Semester Workload Lectures 39 Individual study 78 Solving exercises-homework 33 Course total 150
Student Performance Evaluation	Exams in the end of the semester (mandatory), intermediate exams (optional), assignments of exercises during the semester (optional).

Attached Bibliography

See Eudoxus.