Course: Introduction to Functional Analysis

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Course title Introduction to Functional Analysis
Course code UMB/580
Organizational form of instruction Lecture
Level of course Bachelor
Year of study not specified
Frequency of the course In each academic year, in the winter semester.
Semester Winter
Number of ECTS credits 5
Language of instruction Czech
Status of course Compulsory-optional
Form of instruction Face-to-face
Work placements This is not an internship
Recommended optional programme components None
Lecturer(s)
  • Eisner Jan, Mgr. Dr.
Course content
Sigma algebra, introduction of a Lebesgue measure, measurable sets, measurable functions. Definition of the Lebesgue integral - fundamental properties. Convergence theorems. Metric spaces, normed linear spaces, Banach and Hilbert spaces. Spaces of smooth functions, Lp spaces. Linear operators, continuous operators.

Learning activities and teaching methods
Monologic (reading, lecture, briefing), Demonstration
  • Class attendance - 42 hours per semester
  • Preparation for classes - 42 hours per semester
  • Preparation for exam - 42 hours per semester
Learning outcomes
Students should acquaint themselves with Lebesgue integral and basic functional spaces.
We introduce sigma algebra, outer measure and then Lebesgue measure, measure sets and functions. We define Lebesgue integral and show its basic properties. We work with metric spaces, normed spaces, in particular Banach and Hilbert spaces. As examples we will use spaces of smooth functions, Lp spaces and Sobolev spaces. We will study linear and continuous operators and their properties. We show a power and application of fixed point theorems.
Prerequisites
We assume knowledge of differentail and integral calculus (UMB/564, UMB/565, UMB/566) and Linear algebra UMB/551. It is better to pass before also Linear algebra II UMB/585, where similar topics is studied in spaces of finite dimension.
UMB/CV566
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UMB/CV585
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UMB/566
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UMB/566K
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UMB/585

Assessment methods and criteria
Oral examination

Active participation on lectures, solving of simple homeworks. Final oral exam.
Recommended literature
  • A. N. Kolmogorov, S.V. Fomin: Základy teorie funkcí a funkcionální analýzy. 1. vyd. Praha : SNTL - Nakladatelství technické literatury, 1975..
  • M. Giaquinta, G. Modica: Mathematical Analysis. An Introduction to Functions of Several Variables. Birkhauser Boston, 2009..
  • M. Giaquinta, G. Modica: Mathematical Analysis, Linear and Metric Structures and Continuity. Birkhäuser Boston, 2007..
  • V. Jarník: Diferenciální počet II, Integrální počet II, Academia Praha, 1984..
  • J. Kopáček. Matematika nejen pro fyziky III. Praha matfyzpress, 2007.


Study plans that include the course
Faculty Study plan (Version) Category of Branch/Specialization Recommended year of study Recommended semester
Faculty: Faculty of Science Study plan (Version): Secondary Schools Teacher Training in Mathematics (2012) Category: Pedagogy, teacher training and social care - Recommended year of study:-, Recommended semester: Winter
Faculty: Faculty of Science Study plan (Version): Applied Mathematics (2010) Category: Mathematics courses 3 Recommended year of study:3, Recommended semester: Winter
Faculty: Faculty of Science Study plan (Version): Secondary Schools Teacher Training in Mathematics (1) Category: Pedagogy, teacher training and social care - Recommended year of study:-, Recommended semester: Winter
Faculty: Faculty of Science Study plan (Version): Secondary Schools Teacher Training in Mathematics (1) Category: Pedagogy, teacher training and social care - Recommended year of study:-, Recommended semester: Winter
Faculty: Faculty of Science Study plan (Version): Secondary Schools Teacher Training in Mathematics (1) Category: Pedagogy, teacher training and social care - Recommended year of study:-, Recommended semester: Winter