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Course detail

Functional Analysis I

Course unit code: FSI-SU1
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:
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 % face-to-face, 10 % distance learning
Prerequisites:
Differential and integral calculus, numerical methods, ordinary differential equations.
Co-requisites:
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, PC-DIR, 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, Springer-Verlag, 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:
Course-unit 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 subject-matter 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, Riesz-Fischer theorem, Hilbert space, characteristic property of Euclidian spaces.
4. Functionals - definition and examples, geometric interpretation, convex sets, convex functionals, Hahn-Banach theorem, continuous linear functionals, Hahn-Banach theorem in normed spaces.
5. Adjoint spaces - definition and examples, second adjoint spaces, weak convergence, Banach-Steinhaus 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 self-study using recommended literature.

Type of course unit:

Lecture: 26 hours, optionally
Teacher / Lecturer: prof. Aleksandre Lomtatidze, DrSc.
Syllabus: 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, Riesz-Fischer theorem, Hilbert space, characteristic property of Euclidian spaces.
4. Functionals - definition and examples, geometric interpretation, convex sets, convex functionals, Hahn-Banach theorem, continuous linear functionals, Hahn-Banach theorem in normed spaces.
5. Adjoint spaces - definition and examples, second adjoint spaces, weak convergence, Banach-Steinhaus theorem, weak convergence and bounded sets in adjoint spaces.
seminars: 26 hours, compulsory
Teacher / Lecturer: prof. Aleksandre Lomtatidze, DrSc.
Syllabus: Practising the subject-matter presented at the lectures on particular examples of finite dimensional spaces, spaces of sequences and spaces of continuous and integrable functions.

The study programmes with the given course