M7180 Functional Analysis II

Faculty of Science
Autumn 2011
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)
prof. Alexander Lomtatidze, DrSc. (lecturer)
Guaranteed by
prof. Alexander Lomtatidze, DrSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Tue 17:00–18:50 M6,01011
  • Timetable of Seminar Groups:
M7180/01: Tue 19:00–19:50 M6,01011, A. Lomtatidze
Prerequisites
M6150 Functional Analysis I
Differential and integral calculus, Linear functional analysis I.
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
there are 6 fields of study the course is directly associated with, display
Abstract
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 explain bases of theory of linear operators, spectral theory and theory of operator equations. At the end of the course students should be able to: 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.
Key topics
  • 1. Linear operators. Definition and examples. Bounded and continuous operators. Inverse operators. Adjoint operators. Compact operators.
  • 2. Spectrum. Bases of the spectral analysis. Classification. Spectrum of the compact operator.
  • 3. Operator equations. Fredholm theory. Riesz-Schauder theory. Applications.
  • 4. Lerey-Schauder degree of mapping. Fixed point theorems. Solvabulity of nonlinear equations in Banach spaces.
Study resources and literature
  • Lang, S. Real and Functional Analysis. Third Edition. Springer-Verlag 1993.
  • Dunford, N. - Schwartz, T. Linear operators. Part I: General theory. New York and London: Interscience Publishers. XIV, 1958, 858 p.
  • DRÁBEK, Pavel and Jaroslav MILOTA. Methods of nonlinear analysis : applications to differential equations. Basel: Birkhäuser, 2007, xii, 568. ISBN 9783764381462. info
  • DRÁBEK, Pavel and Jaroslav MILOTA. Lectures on nonlinear analysis. 1. vyd. Plzeň: Vydavatelský servis, 2004, xi, 353. ISBN 8086843009. info
  • ZEIDLER, Eberhard. Applied functional analysis : main principles and their applications. New York: Springer-Verlag, 1995, xvi, 404. ISBN 0387944222. info
  • KOLMOGOROV, Andrej Nikolajevič and Sergej Vasil‘jevič FOMIN. Základy teorie funkcí a funkcionální analýzy. Translated by Vladimír Doležal - Zdeněk Tichý. Vyd. 1. Praha: SNTL - Nakladatelství technické literatury, 1975, 581 s. info
Approaches, practices, and methods used in teaching
lectures and class exercises
Method of verifying learning outcomes and course completion requirements
Teaching: lecture 2 hours a week, seminar 1 hour a week. Examination: written and oral.
Language of instruction
Czech
Teacher's information
Předmět je ukončen zkouškou, která má dvě části - ústní a písemnou. Požadavky na úspěšné zakončení předmětu: Písemná část zkoušky představuje zpracování zadaného tématu jako přípravu k části ústní. V průběhu ústní zkoušky je požadováno pochopení zavedených pojmů, porozumění vyloženým větám a schopnost jejich formulace. Je vyžadována znalost jednodušších důkazů a myšlenkových postupů složitějších důkazů.
Further comments (probably available only in Czech)
The course is taught once in two years.
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 1999, Autumn 2001, Autumn 2003, Autumn 2005, Autumn 2007, Autumn 2009, Autumn 2013, Autumn 2015, autumn 2017, Autumn 2019, autumn 2021, Autumn 2023, Autumn 2025.
  • Enrolment Statistics (Autumn 2011, recent)
  • Permalink: https://is.muni.cz/course/sci/autumn2011/M7180