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Course, academic year 2022/2023
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Introduction to Functional Analysis (O) - NMMA931
Title: Úvod do funkcionální analýzy (O)
Guaranteed by: Department of Mathematical Analysis (32-KMA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2022
Semester: winter
E-Credits: 8
Hours per week, examination: winter s.:4/2, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
Virtual mobility / capacity: no
State of the course: taught
Language: Czech
Teaching methods: full-time
Is provided by: NMMA331
Guarantor: prof. RNDr. Ondřej Kalenda, Ph.D., DSc.
Class: Informatika Mgr. - Diskrétní modely a algoritmy
Classification: Mathematics > Functional Analysis
Incompatibility : NMMA331, NRFA006, NRFA106
Interchangeability : NMMA331, NRFA106
Is interchangeable with: NRFA106
Annotation -
Last update: G_M (16.05.2012)
An introductory course in functional analysis. Not equivalent to the course NMMA331.
Course completion requirements -
Last update: prof. RNDr. Ondřej Kalenda, Ph.D., DSc. (12.09.2022)

The course is taught in Czech, so see the Czech version.

Literature - Czech
Last update: G_M (27.04.2012)

Habala, Hájek, Zizler, Banach Spaces I, II (skripta, MATFYZpress 1997)

M. Katětov a J. Jelínek, Úvod do funkcionální analýzy (skripta, SPN Praha 1968)

J. Lukeš, Uvod do funkcionální analýzy (skripta, Karolinum Praha, 2005)

J. Lukeš, Zápisky z funkcionální analýzy (skripta, Karolinum Praha 1998, 2002, 2003)

J. Lukeš a J. Malý, Míra a integrál (skripta, Univerzita Karlova, 1993, 2002 - anglické vydání 1995, 2005)

L. Mišík, Funkcionálna analýza (Alfa Bratislava, 1989)

K. Najzar, Funkcionální analýza (skripta, SPN Praha 1988)

I. Netuka a J. Veselý, Příklady z funkcionální analýzy (skripta MFF UK 1972)

P. Quittner, Funkcionálna analýza v príkladoch (Veda, SAV Bratislava 1990)

W. Rudin, Analýza v reálném a komplexním oboru (Academia Praha 1977, 2003)

W. Rudin, Functional analysis (Mc Graw Hill 1973 - ruský překlad 1975)

J. Stará, Příklady z matematické analýzy IV: Funkcionální analýza (skripta, SPN Praha 1975)

A.E. Taylor, Úvod do funkcionální analýzy (Academia Praha 1973)

Requirements to the exam -
Last update: prof. RNDr. Ondřej Kalenda, Ph.D., DSc. (12.09.2022)

The course is taught in Czech, so see the Czech version.

Syllabus -
Last update: prof. RNDr. Ondřej Kalenda, Ph.D., DSc. (12.09.2022)
1. Banach and Hilbert spaces
normed spaces, spaces with inner product, examples of Banach spaces

continuous linear mappings - characterization, norm, space of operators

convergence of series in Banach spaces

Hilbert spaces - orthonormal systems, orthonormal basis, Riesz-Fischer etc.

finite-dimensional vs infinite-dimensional spaces

real spaces vs. complex spaces

2. Duality and Hahn-Banach theorem
Hahn-Banach theorem and its consequences

separation of convex sets

canonical embedding into second dual and reflexive spaces

representation of dual spaces to classical Banach spaces

wead (and weak*) convergence of sequences (definition, comparision, examples, characterization in classical spaces)

choice of weakly convergent subsequences in reflexive spaces (and weak*-converent subsequences in duals of separable spaces)

3. Operators on Banach spaces
Principle of uniform boundedness, Banach-Steinhaus and consequences

Open mapping theorem and Closed graph theorem

Quotient, projection, complementability

Dual operators, duality of subspaces and quotients

Adjoint operators between Hilbert spaces

Spectrum of operators

Compact operators - definition, properties, structure of the spectrum

Selfadjoint compact operators on Hilbert space

4. Fourier transformation
Definition and properties of Fourier transformation on L_1

Schwartz space and Fourier transformation on it

Inverse theorem

Plancherel transformation on L_2

Entry requirements -
Last update: prof. RNDr. Ondřej Kalenda, Ph.D., DSc. (12.09.2022)

The lecture requires previous fair knowledge from mathematical analysis (Mathematical analysis 1-3, metric spaces from Mathematical analysis 4), linear algebra (mainly vector spaces and linear mappings, with emphasis on infinite-dimensional spaces), and Theory of measure and integral.

 
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