Course detail

Functional Analysis I

FSI-SU1Acad. year: 2011/2012

The course deals with basic topics of the functional analysis and their illustration on particular metric, linear normed and unitary spaces. Lebesgue measure, Lebesgue integral and spaces of integrable functions are 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. RNDr. Jan Franců, CSc.

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

Teaching methods depend on the type of course unit as specified in the article 7 of BUT Rules for Studies and Examinations.

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

Not applicable.

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)

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. RNDr. Jan Franců, CSc.

Syllabus

1 Metric spaces, definitions and examples. Open and closed sets.
2 Convergence. Closure, interior and boundary of a set, examples in finite dimension.
3 Separable, totaly bounded sets, nets, spaces of sequentions.
4 Compact sets and complete spaces. Spaces of continuous functions.
5 Theorem on completion. Banach theorem on fixed point and its applications.
6 Measurable sets and measure in R^1
7 Mesurable functions and Lebesgue integral.
8 Theorems on limit passages.
9 Lebesgue measure and integral in R^n.
10 Spaces of integrable functions, inequalities.
11 Linear and normed linear spaces, linear functionals and their norm.
12 Scalar product and Hilbert space. Abstract Fourier series.
13 Reserve.

Exercise

26 hod., compulsory

Teacher / Lecturer

prof. RNDr. Jan Franců, CSc.

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.