MU20025 Functional Analysis

Mathematical Institute in Opava
Winter 2021
Extent and Intensity
2/2/0. 6 credit(s). Type of Completion: zk (examination).
doc. RNDr. Jana Kopfová, Ph.D. (lecturer)
RNDr. Petr Vojčák, Ph.D. (seminar tutor)
Guaranteed by
prof. RNDr. Miroslav Engliš, DrSc.
Mathematical Institute in Opava
Wed 9:45–11:20 RZ
  • Timetable of Seminar Groups:
MU20025/01: Mon 9:45–11:20 R2, 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
Course objectives
The goal of the course is to get students acquainted with basic linear functional analysis including the theory of distributions (generalized functions).
  • 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.
    required 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
    recommended literature
  • W. Rudin. Functional Analysis. 1991. info
    not specified
  • L. Kosmák, R. Potůček. Metrické prostory. Praha, 2011. info
  • I. M. Gelfand, G. E. Shilov. Generalized functions: Volume II, Spaces of Fundamental and Generalized Functions. 1968. info
  • I.M.Gelfand, G. E. Shilov. Generalized Functions: Volume I, Properties and operations. 1964. info
Language of instruction
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.
The course is also listed under the following terms Winter 2019, Winter 2020.
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