M6150 Linear Functional Analysis I

Faculty of Science
Spring 2008 - for the purpose of the accreditation
Extent and Intensity
2/1/0. 3 credit(s) (fasci plus compl plus > 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
Contact Person: prof. Alexander Lomtatidze, DrSc.
Prerequisites (in Czech)
M3100 Mathematical Analysis III && M4170 Measure and Integral
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
Classification of topological vector spaces (normed spaces, Hilbert spaces, metric vector spaces, locally convex spaces) Linear bounded operators in normed spaces (Hahn-Banach theorem, Banach-Steinhaus theorem, open mapping theorem, closed graph theorem) Dual spaces and operators (representation of linaer bounded functionals, reflexivity, weak convergence, completeness and compactness, basic spectral theory Compact operators (compact sets in normed spaces, compact operators, Fredholm theorems, spectral theorems)
Syllabus (in Czech)
  • 1. Metrický prostor. Definice, příklady. Podmnožiny, klasifikace bodů. Konvergence. Úplnost, kompaktnost, spočetná kompaktnost, kompaktnost v některých prostorech. 2. Lineární prostor. Definice, příklady. Normovaný prostor. Unitární prostor. Besselova nerovnost. Rieszova-Fischerova věta. Hilbertův prostor. Charakteristická vlastnost unitárních prostorů. 3. Funkcionály. Definice, příklady. Geometrický význam lineárního funkcionálu. Konvexní množiny a konvexní funkcionály. Hahnova-Banachova věta a její aplikace. Spojité lineární funkcionály. Hahnova-Banachova věta v normovaném prostoru. 4. Adjungovaný prostor. Definice, příklady. Úplnost. Prostor adjungované k Hilbertovému prostoru. Druhý adjungovaný prostor. Banachova-Steinhausova věta, slabá konvergence. 5. Slabá konvergence a ohraničené množiny v adjungovaném prostoru.
Literature
  • Lang, S. Real and Functional Analysis. Third Edition. Springer-Verlag 1993.
  • 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
Language of instruction
Czech
Further Comments
The course is taught annually.
The course is taught: every week.
Listed among pre-requisites of other courses
The course is also listed under the following terms 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 2012 - acreditation, 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.