1. SylabUZ
2. Wydział Nauk Ścisłych i Przyrodniczych
3. semestr zimowy 2024/2025
4. WMIiE - oferta ERASMUS - Program Erasmus
5. Functional Analysis

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Functional Analysis - opis przedmiotu

Cel przedmiotu

The aim of the course is to acquaint students with basic properties of Banach and Hilbert spaces as well as with basis of the theory of linear operators on Banach spaces.

Wymagania wstępne

Zapisz zmiany

It is assumed that students know basis of set theory, metric topology, linear algebra, mathematical analysis and elements of measure and Lebesgue integral theories.

Zakres tematyczny

Lecture

Normed and Banach spaces

1. Normed and Banach spaces. Basic definitions and properties. Examples of sequence an function Banach spaces.  (2 hours)
2. Series in normed spaces. Definitions and examples.  (1 hour)
3. Cartesian product of normed spaces. Completion of a normed space.  (2 hours)
4. Finite dimensional normed spaces. Completeness of finite dimensional spaces. Compactness of sets in finite dimensional spaces. Riesz's theorem.  (3 godz.)

Bounded linear operators on normed spaces

1. Basic properties of bounded linear operators. Examples of bounded linear operators on sequence and function Banach spaces (2 hours)
2. Norm of a bounded linear operator. The space of bounded linear operators. Dual space of a normed space.   (2 hours)
3. Compact linear operators on Banach spaces  (2 hours)
4. Banach-Steinhaus theorem and its applications  (2 hours)
5. Banach's inverse mapping theorem and closed graph theorem.  (2 hours)
6. Hahn-Banach theorem and its applications  (2 hours)
7. General form of continuous linear functionals over classical sequence Banach spaces  (2 hours)

Hibert spaces

1. Inner product spaces and Hilbert spaces - basic definitions and properties. Examples.  (2 hours)
2. Orthogonal projection theorem in Hilbert spaces and its applications.  (2 hours)
3. General form of continuous linear functionals over Hilbert spaces.  (1 hour)
4. Orthogonal systems in Hilbert spaces. Fourier series in Hilbert spaces.  (3 hours)

 

Classes

Normed and Banach spaces

1. Examples of sequence and function linear spaces. Basic properties. Hölder and Minkowski inequalities.  (3 hours)
2. Examining  norm conditions on sequence and function spaces.  Proving completeness of classical sequence and function normed spaces.  (3 hours)
3. Calculation of the norm of elements in sequence and function spaces.  (3 hours)
4. Comparing norms in normed spaces.  (1 hour)
5. Colloquium  (2 hours)

Bounded linear operators on normed spaces

1. Examining linearity and boundedness of functionals and operators defined on sequence and function normed spaces.  (3 hours)
2. Calculation of the norm of linear functionals on sequence and function spaces.  (3 hours)

Hilbert spaces

1. Examples of Hilbert. Basic properties.  (2 hours)
2. Examining conditions of inner product in sequence and function spaces.  (2 hours)
3. Testing geometric and topological properties of Hilbert spaces.  (4 hours)
4. Examining orthogonal systems in Hilbert spaces.  (2 hours)
5. Colloquium  (2 hours)

Metody kształcenia

Conventional (traditional) lecture. Classes (auditorium), solving exercises and problems.

Efekty uczenia się i metody weryfikacji osiągania efektów uczenia się

Opis efektu	Symbole efektów	Metody weryfikacji	Forma zajęć

Warunki zaliczenia

The course completion grade consists of the classes grades (40%) and examination grades (60%). A positive grade of the classes is required to sit for the exam.  A positive grade of the examination is required to credit for the course.

Literatura podstawowa

1. V.L. Hansen, Functional Analysis: Entering Hilbert Spaces, Second Edition, World Scientifing, Singapore, 2016.
2. M.V.Markin, Elementary Functional Analysis, De Gruyter, Berlin/Boston, 2018.     
3. O.M.Shalit, A First Course in Functional Analysis, CRC Press, Boca Raton, 2017                                                              .

Literatura uzupełniająca

1. B Choudhary, Sudarsand Nanda, Functional Analysis with Applications, New Age International (P) Limited, Publishers, New Delhi,  2015 (ebook).
2. K. Saxe, Beginning Functional Analysis, Undergraduate Texts in Mathematics, Springer, New York, 2002.

Uwagi

Zmodyfikowane przez prof. dr hab. Witold Jarczyk (ostatnia modyfikacja: 25-04-2024 14:53)