714-0522/01 – Functional Analysis (FA)

Gurantor departmentDepartment of Mathematics and Descriptive GeometryCredits4
Subject guarantordoc. RNDr. Jaroslav Vlček, CSc.Subject version guarantordoc. RNDr. Jaroslav Vlček, CSc.
Study levelundergraduate or graduateRequirementChoice-compulsory
Year2Semesterwinter
Study languageCzech
Year of introduction1998/1999Year of cancellation2015/2016
Intended for the facultiesHGFIntended for study typesFollow-up Master
Instruction secured by
LoginNameTuitorTeacher giving lectures
VLC20 doc. RNDr. Jaroslav Vlček, CSc.
Extent of instruction for forms of study
Form of studyWay of compl.Extent
Full-time Credit and Examination 2+2
Combined Credit and Examination 12+0

Subject aims expressed by acquired skills and competences

- an obtaining of basic orientation in functional spaces, - skilled use of functional analysis theory in formulation of applied problems.

Teaching methods

Lectures
Individual consultations
Tutorials

Summary

Metric spaces, linear spaces, normed linear spaces, orthogonality. Cauchy's sequences, completeness. Banach and Hilbert spaces. Hilbertovy a Banachovy prostory. Functionals and operators in Hilbert space. Schauder's theorem.

Compulsory literature:

REKTORYS, K.: Variational Methods in Mathematics, Science and Engineering. SNTL, Praha, 1980

Recommended literature:

E. Zeidler: Applied Functional Analysis. Springer-Verlag, New York, 1995

Way of continuous check of knowledge in the course of semester

Cvičení (10-30 bodů): - aktivní účast ve výuce (20% neúčasti tolerováno) ... 10 b. - testy ... 20 b. Zkouška (0-70 b.): - praktická (písemná část) ... 50 b. - teoretická část ... 20 b. 1. Metrický prostor. Základní metrické a topologické pojmy. 2. Separabilita a kompaktnost. Cauchyovské posloupnosti. Úplnost. 3. Lineární prostor. Definice a základní vlastnosti. Konvexnost. 4. Normovaný lineární prostor. Norma. Metrika indukovaná normou. 5. Banachův prostor. Věta o pevném bodě. 6. Unitární prostor. Skalární součin. Metrika indukovaná skalárním součinem. 7. Hilbertův prostor. Rieszova věta o reprezentaci. 8. Ortogonální systémy. Příklady. 9. Lineární funkcionály a operátory v Hilbertově prostoru.

E-learning

Další požadavky na studenta

The other demands are not prescribed.

Prerequisities

Subject has no prerequisities.

Co-requisities

Subject has no co-requisities.

Subject syllabus:

1. Metric spaces. Separability and compactness. Cauchy sequences. Completness. 2. Linear space. Definition and basic properties. Convexity. Examples of linear spaces. 3. Norm. Metric induced by norm. Banach space. 4. Unitary spaces. Scalar product. Metric indued by scalar product. Hilbert space. Riesz theorem. 5. Ortogonal systems. 6. Linear operstors in Hilbert space.

Conditions for subject completion

Full-time form (validity from: 2010/2011 Summer semester, validity until: 2012/2013 Winter semester)
Task nameType of taskMax. number of points
(act. for subtasks)
Min. number of points
Exercises evaluation and Examination Credit and Examination 100 (100) 51
        Exercises evaluation Credit 30  10
        Examination Examination 70  0
Mandatory attendence parzicipation:

Show history

Occurrence in study plans

Academic yearProgrammeField of studySpec.FormStudy language Tut. centreYearWSType of duty
2012/2013 (N2102) Mineral Raw Materials (3911T001) Applied Physics of Materials P Czech Ostrava 2 Choice-compulsory study plan
2011/2012 (N2102) Mineral Raw Materials (3911T001) Applied Physics of Materials P Czech Ostrava 2 Choice-compulsory study plan
2010/2011 (N2102) Mineral Raw Materials (3911T001) Applied Physics of Materials P Czech Ostrava 2 Choice-compulsory study plan
2009/2010 (N2102) Mineral Raw Materials (3911T001) Applied Physics of Materials P Czech Ostrava 2 Choice-compulsory study plan
2005/2006 (M2102) Mineral Raw Materials (3911T001) Applied Physics of Materials P Czech Ostrava 4 Choice-compulsory study plan
2004/2005 (M2102) Mineral Raw Materials (3911T001) Applied Physics of Materials P Czech Ostrava 4 Choice-compulsory study plan
2003/2004 (M2102) Mineral Raw Materials (3911T001) Applied Physics of Materials P Czech Ostrava 4 Choice-compulsory study plan
2002/2003 (M2102) Mineral Raw Materials (3911T001) Applied Physics of Materials P Czech Ostrava 4 Choice-compulsory study plan
2001/2002 (M2102) Mineral Raw Materials (3911T001) Applied Physics of Materials P Czech Ostrava 4 Choice-compulsory study plan

Occurrence in special blocks

Block nameAcademic yearForm of studyStudy language YearWSType of blockBlock owner