#
PřF:M6150 Functional Analysis I - Course Information

## M6150 Functional Analysis I

**Faculty of Science**

Spring 2020

**Extent and Intensity**- 2/1/0. 3 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Type of Completion: zk (examination).
**Teacher(s)**- Mgr. Peter Šepitka, Ph.D. (lecturer)
**Guaranteed by**- doc. RNDr. Michal Veselý, Ph.D.

Department of Mathematics and Statistics - Departments - Faculty of Science

Supplier department: Department of Mathematics and Statistics - Departments - Faculty of Science **Timetable of Seminar Groups**- M6150/01: No timetable has been entered into IS.
*P. Šepitka* **Prerequisites**-
**M3100**Mathematical Analysis III

Mathematical analysis: Differential calculus of functions of one and several variables, integral calculus, sequences and series of numbers and functions. Linear algebra: Systems of linear equations, determinants, linear spaces, linear transformation and matrices, canonical form of a matrix. **Course Enrolment Limitations**- The course is also offered to the students of the fields other than those the course is directly associated with.
**fields of study / plans the course is directly associated with**- Finance Mathematics (programme PřF, N-MA)
- Mathematics (programme PřF, B-MA)
- Statistics and Data Analysis (programme PřF, B-MA)

**Course objectives**- Functional analysis belongs to fundamental parts of university courses in mathematics. It is utilized by a number of other courses and in many applications. The aim of the course is to introduce the bases of the linear functional analysis, namely the theory of infinite dimensional vector spaces and their duals.
**Learning outcomes**- At the end of the course, students should be able: to define and interpret the basic notions used in the fields mentioned above; to formulate relevant mathematical theorems and statements and to explain methods of their proofs; to use effective techniques utilized in these subject areas; to apply acquired pieces of knowledge for the solution of specific problems; to analyse selected problems from the topics of the course.
**Syllabus**- 0. Metric spaces: Basic examples. Closed and open sets. Limits of sequences. Maps of metric spaces. Complete metric spaces. Compact spaces.
- 1. Normed linear spaces, Hilbert spaces: Basic differences between finite and infinite dimensional linear spaces. Spaces of functions and sequences. Orthogonality in Hilbert spaces. General Fourier series.
- 2. Linear functionals: norm, continuity, boundedness, invertibility. The Hahn-Banach theorem and its consequences.
- 3. Dual (adjoint) spaces: Dual spaces to functions and sequences spaces. Week convergence and reflexivity. The Banach-Steinhaus theorem and its consequences.

**Literature***A guide to functional analysis*. Edited by Steven G. Krantz. Washington, D.C.: Mathematical Association of America, 2013. xii, 137 p. ISBN 9781614442134. info- DOŠLÁ, Zuzana and Ondřej DOŠLÝ.
*Metrické prostory : teorie a příklady*. 3. vyd. Brno: Masarykova univerzita, 2006. viii, 90. ISBN 8021041609. info - NAJZAR, Karel.
*Funkcionální analýza*. 1. vyd. Praha: Státní pedagogické nakladatelství, 1975. 183 s. info - KOLMOGOROV, A. N. and S. V. FOMIN.
*Základy teorie funkcí a funkcionální analýzy*. 1. vyd. Praha: SNTL - Nakladatelství technické literatury, 1975. 581 s. info

*recommended literature***Teaching methods**- Lectures and class exercises
**Assessment methods**- The final oral exam for 20 points. For successfull examination (the grade at least E), the student needs 10 points or more.
**Language of instruction**- Czech
**Follow-Up Courses****Further comments (probably available only in Czech)**- Study Materials

The course is taught annually. **Teacher's information**- The lessons are usually in Czech or in English as needed, and the
relevant terminology is always given with English equivalents.
The target skills of the study include the ability to use the English language passively and actively in their own expertise and also in potential areas of application of mathematics.

Assessment in all cases may be in Czech and English, at the student's choice.

- Enrolment Statistics (recent)

- Permalink: https://is.muni.cz/course/sci/spring2020/M6150