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MU:MU20025 Functional Analysis - Course Information

## MU20025 Functional Analysis

**Mathematical Institute in Opava**

Winter 2020

**Extent and Intensity**- 2/2/0. 6 credit(s). Type of Completion: zk (examination).
**Teacher(s)**- prof. RNDr. Miroslav Engliš, DrSc. (lecturer)

RNDr. Petr Vojčák, Ph.D. (seminar tutor) **Guaranteed by**- prof. RNDr. Miroslav Engliš, DrSc.

Mathematical Institute in Opava **Timetable**- Tue 14:45–16:20 19
- Timetable of Seminar Groups:

*P. Vojčák* **Prerequisites**(in Czech)-
**MU20004**Mathematical Analysis IV &&**MU20006**Algebra II && TYP_STUDIA ( B ) **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**- Mathematical Methods and Modelling (programme MU, Bc-M)
- General Mathematics (programme MU, Bc-M)

**Course objectives**- The goal of the course is to get students acquainted with basic linear functional analysis including the theory of distributions (generalized functions).
**Syllabus**- 1. Metric spaces: metric and its definition, examples, metric derived from a norm, inner product spaces; topology of metric spaces: open sets, convergence, continuous mappings, compactness, separability, connectedness; uniform continuity and completeness, Baire's category theorem.

2. Hilbert and Banach spaces: Hilbert space, definition, examples; closest point theorem and Riesz representation theorem; spectral theory for compact operators, Fredholm's integral equations; Banach spaces: definition and examples; separation of convex sets and Hahn-Banach theorem; dual spaces, weak convergence and Banach-Steinhaus theorem; open mapping theorem and closed graph theorem.

3. Theory of distributions: vector space topology generated by a system of seminorms and its fundamental properties, strict inductive limit of locally convex topological vector spaces; gauge functional and Kolmogorov's criterion, countably normed spaces, examples of locally convex function spaces; dual spaces and distributions, operations with distributions; tempered distributions and Fourier transform.

- 1. Metric spaces: metric and its definition, examples, metric derived from a norm, inner product spaces; topology of metric spaces: open sets, convergence, continuous mappings, compactness, separability, connectedness; uniform continuity and completeness, Baire's category theorem.
**Literature**- B. Simon.
*Real Analysis: A Comprehensive Course in Analysis, Part I*. 2015. info - J. Muscat.
*Functional Analysis*. 2014. info - I. Netuka.
*Základy moderní analýzy*. 2014. info - H. W. Alt.
*Linear Functional Analysis: An Application-Oriented Introduction*. 2012. info - B.P. Rynne, M.A. Youngson.
*Linear Functional Analysis*. 2000. info

*required literature*- W. Rudin.
*Functional Analysis*. 1991. info

*recommended literature*- B. Simon.
**Language of instruction**- Czech
**Further comments (probably available only in Czech)**- Study Materials

The course can also be completed outside the examination period. **Teacher's information**- To obtain course credits it is necessary to solve three problems assigned to the student by the instructor. Final exam consists of two theoretical questions.

- Enrolment Statistics (Winter 2020, recent)
- Permalink: https://is.slu.cz/course/sumu/winter2020/MU20025