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
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
    recommended 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
    not specified
  • ZEIDLER, Eberhard. Applied functional analysis : main principles and their applications. New York: Springer-Verlag, 1995. xvi, 404. ISBN 0387944222. info
  • CONWAY, John B. A course in functional analysis. 2-nd ed. New York: Springer - Verlag, 1990. xvi, 399. ISBN 0387972455. info
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.

The course is also listed under the following terms Spring 2008 - for the purpose of the accreditation, Spring 2011 - only for the accreditation, Spring 2000, Spring 2001, Spring 2002, Spring 2003, Spring 2004, Spring 2005, Spring 2006, Spring 2007, Spring 2008, Spring 2009, Spring 2010, Spring 2011, Spring 2012, spring 2012 - acreditation, Spring 2013, Spring 2014, Spring 2015, Spring 2016, Spring 2017, spring 2018, Spring 2019.
  • Enrolment Statistics (recent)
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