Course unit code: 
FSISU1 
Academic year: 
2016/2017 
Type of course unit: 
compulsory 
Level of course unit: 
Bachelor's (1st cycle) 
Year of study: 
2 
Semester: 
summer 
Number of ECTS credits: 
5 
Learning outcomes of the course unit:
Knowledge of basic topics of the metric, linear, normed and unitary spaces, Lebesgue integral
and ability to apply this knowledge in practice.


Mode of delivery:
90 % facetoface, 10 % distance learning


Prerequisites:
Differential and integral calculus, numerical methods, ordinary differential equations.


Corequisites:
Not applicable.


Recommended optional programme components:
Not applicable.


Course contents (annotation):
The course deals with basic topics of the functional analysis and their illustration on particular metric, linear normed and unitary spaces. Lebesgue measure and Lebesgue integral are also introduced. The results are applied to solving of problems of mathematical and numerical analysis.


Recommended or required reading:
A. N. Kolmogorov, S. V. Fomin: Základy teorie funkcí a funkcionální analýzy, SNTL, Praha 1975. A.Ženíšek: Lebesgueův integrál a základy funkcionální analýzy, skripta FSI VUT, PCDIR, Brno 1999. J. Lukeš: Úvod do funkcionální analýzy, Učební texty UK v Praze, Nakladatelství Karolinum, Praha 2005 I. P. Natanson: Teorija funkcij veščestvennoj peremennoj, Moskva E. Čech: Bodové množiny, Academia, Praha 1974 S. Lang: Real and Functional Analysis, SpringerVerlag, New York, 1993.


Planned learning activities and teaching methods:
The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.


Assesment methods and criteria linked to learning outcomes:
Courseunit credit is awarded on condition of having attended the seminars actively and passed the control test.
Examination has a practical and a theoretical part. In the practical part student has to illustrate the given tasks on particular examples.
Theoretical part includes questions related to the subjectmatter presented at the lectures.


Language of instruction:
Czech


Work placements:
Not applicable.


Course curriculum:
1. Metric spaces  definition and examples, classification of subsets, separability, covergence, completness, compactness, compactness of sets in particular spaces.
2. Measura theory  Lebesgue measure, measurable functions, Lebesgue integral, Lebesgue dominant theorem.
3. Linear spaces  definition and examples, normed space, Euclidian space, Bessel inequality, RieszFischer theorem, Hilbert space, characteristic property of Euclidian spaces.
4. Functionals  definition and examples, geometric interpretation, convex sets, convex functionals, HahnBanach theorem, continuous linear functionals, HahnBanach theorem in normed spaces.
5. Adjoint spaces  definition and examples, second adjoint spaces, weak convergence, BanachSteinhaus theorem, weak convergence and bounded sets in adjoint spaces.


Aims:
The aim of the course is to familiarise students with basic topics and procedures of the functional analysis used in other mathematical subjects.


Specification of controlled education, way of implementation and compensation for absences:
Absence has to be made up by selfstudy using recommended literature.

