Course detail

# Functional Analysis I

FSI-SU1CompulsoryBachelor's (1st cycle)Acad. year: 2016/2017Summer semester2. year of study5 credits

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.

Supervisor

Department

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.

Recommended optional programme components

Not applicable.

Recommended or required reading

F. Burk, Lebesgue measure and integration: An introduction, Wiley 1998.

C. Costara, D. Popa, Exercises in functional analysis, Kluwer 2003.

Z. Došlá, O. Došlý, Metrické prostory: teorie a příklady, PřF MU Brno 2006.

J. Franců, Funkcionální analýza 1, FSI VUT 2014.

D. H. Griffel, Applied functional analysis, Dover 2002.

A. N. Kolmogorov, S. V. Fomin: Základy teorie funkcí a funkcionální analýzy, SNTL, Praha 1975.

E. Kreyszig, Introductory functional analysis with applications, J. Wiley 1978.

J. Lukeš, Zápisky z funkcionální analýzy, Karolinum 1998.

B. Rynne, M. Youngson, Linear functional analysis, Springer 2008.

K. Saxe, Beginning functional analysis, Springer 2002.

A. E. Taylor, Úvod do funkcionální analýzy, Academia, Praha 1973.

A. Torchinsky, Problems in real and functional analysis, American Mathematical Society 2015.

E. Zeidler, Applied functional analysis: Main principles and their applications, Springer, 1995.

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

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

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.