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Introduction to Functional Analysis - NMMA331

Title:	Úvod do funkcionální analýzy
Guaranteed by:	Department of Mathematical Analysis (32-KMA)
Faculty:	Faculty of Mathematics and Physics
Actual:	from 2025
Semester:	winter
E-Credits:	8
Hours per week, examination:	winter s.:4/2, C+Ex [HT]
Capacity:	unlimited
Min. number of students:	unlimited
4EU+:	no
Virtual mobility / capacity:	no
State of the course:	taught
Language:	Czech
Teaching methods:	full-time

Guarantor:	prof. RNDr. Ondřej Kalenda, Ph.D., DSc.
Teacher(s):	prof. RNDr. Ondřej Kalenda, Ph.D., DSc.
Class:	DS, matematické a počítačové modelování M Bc. OM M Bc. OM > Zaměření MA M Bc. OM > Zaměření NUMMOD M Bc. OM > Povinně volitelné
Classification:	Mathematics > Functional Analysis
Comes under:	Doporučené přednášky 2/2
Pre-requisite :	{One 2nd year Analysis course}, {NMMA205 v NMMA203}
Incompatibility :	NMMO302, NRFA006
Interchangeability :	NRFA006
Is incompatible with:	NMMA931, NMMA342, NMMO302
Is pre-requisite for:	NMMA349, NMMA351, NMNM351, NMNM349
Is interchangeable with:	NMMA342, NRFA006, NMMA931

Opinion survey results Examination dates WS schedule Noticeboard

Annotation -

Basic course of linear functional analysis for bachelor's program in General Mathematics. Recommended for specializations Mathematical Analysis and Mathematical Modelling and Numerical Analysis..

Last update: G_M (16.05.2012)

Course completion requirements -

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

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

Literature - Czech

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 1991 - 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)

Last update: Spurný Jiří, prof. RNDr., Ph.D., DSc. (07.09.2012)

Requirements to the exam -

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

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

Syllabus -

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

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

Entry requirements -

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.

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