M6150 Functional Analysis I

Faculty of Science
spring 2012 - acreditation

The information about the term spring 2012 - acreditation is not made public

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
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Prerequisites
M3100 Mathematical Analysis III && M4170 Measure and Integral
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. 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.
Syllabus
  • 1. Metric spaces. Definition and examples. Classification of subsets, open and closed subsets. Convergence in metric spaces. Complete metric spaces. Compactness criteria in metric spaces. 2. Linear spaces. Definition and examples. Normed spaces. Inner product. Bessel inequality. Riesz-Fischer theorem. Hilbert spaces. Characteristic properties of vector spaces with inner product. 3. Functionals. Definition and examples. Geometry of linear functionals.Convex sets and convex functionals. Hahn-Banach theorem and its applications. Bounded functionals. Hahn-Banach theorem in normed spaces. 4. Dual spaces. Definition and examples. Completeness. Higher duals. Banach-Steinhaus theorem. Weak convergence. 5. Weak convergence and bounded sets in dual spaces.
Literature
  • Lang, S. Real and Functional Analysis. Third Edition. Springer-Verlag 1993.
  • 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
Teaching methods
lectures and class exercises
Assessment methods
Teaching: lecture 2 hours a week, seminar 1 hours a week. Examination: written and oral.
Language of instruction
Czech
Follow-Up Courses
Further comments (probably available only in Czech)
The course is taught annually.
The course is taught: every week.
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 2013, Spring 2014, Spring 2015, Spring 2016, Spring 2017, spring 2018, Spring 2019, Spring 2020, Spring 2021, Spring 2022, Spring 2023, Spring 2024, Spring 2025.