Course detail

Functional Analysis I

FSI-SU1Acad. year: 2014/2015

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.

Language of instruction

Czech

Number of ECTS credits

6

Mode of study

Not applicable.

Guarantor

prof. Aleksandre Lomtatidze, DrSc.

Department

Institute of Mathematics (ÚM)

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.

Prerequisites

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

Co-requisites

Not applicable.

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.

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.

Work placements

Not applicable.

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.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

A. N. Kolmogorov, S. V. Fomin: Základy teorie funkcí a funkcionální analýzy, SNTL, Praha 1975. (CS)
C. Costara, D. Popa, Exercises in functional analysis, Kluwer 2003. (EN)
F. Burk, Lebesgue measure and integration: An introduction, Wiley 1998. (EN)
J. Franců, Funkcionální analýza 1, FSI VUT 2014. (CS)

Recommended reading

Not applicable.

Classification of course in study plans

  • Programme B3901-3 Bachelor's

    branch B-MAI , 2 year of study, summer semester, compulsory

Type of course unit

 

Lecture

26 hod., 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.

Exercise

26 hod., 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.