Course Syllabus for

Functional Analysis in System Theory
Funktionalanalys i systemteori

FMA003F, 9 credits

Valid from: Autumn 2013
Decided by: FN1/Anders Gustafsson
Date of establishment: 2014-06-09

General Information

Division: Mathematics
Course type: Third-cycle course
Teaching language: English


System theory uses often linear models to describe and optimize dynamic processes. The main goal of the course is to introduce linear systems as abstract linear operators and to give knowledge about basic notions and methods in functional analysis that are used to study and solve optimization problems for such operators i normed spaces. The course develops also an ability to mathematical abstraction that makes it easier to see similarities between different problems, and is suitable for diverse applications, such as control theory, signal processing etc.


Knowledge and Understanding

For a passing grade the doctoral student must

Competences and Skills

For a passing grade the doctoral student must

Course Contents

Normed vector spaces, Banach/Hilbert spaces. Linear operators, adjoint and invers operator. Linear systems as operators, adjoint systems, stability. Quadratic optimization-problems for linear systems. Causal and time-invariant systems, Hankel/Toeplitz operators, transfer function. Topological vector spaces, linear functionals, dual space. Weak topologies. Optimization in Banach/Hilbert spaces. Min-max theorem and duality. Minimum norm theorems. Nehari theorem and other extremal problems in Hardy spaces. Hahn-Banach theorem and separation of convex sets. Convex analysis in normed spaces.

Course Literature

The course literature is a compilation from several books in functional analysis and optimization (available) as well as the lecture notes

Instruction Details

Types of instruction: Lectures, exercises

Examination Details

Examination formats: Written exam, oral exam, written assignments. Weekly hand-in problems or take-home exam Students should take an active role in the weekly exercise sessions
Grading scale: Failed, pass

Admission Details

Assumed prior knowledge: Linear algebra, Multidimensional analysis, Complex Function theory, Systems and Transforms

Course Occasion Information

Contact and Other Information

Course coordinator: Andrey Ghulchak <>

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