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Mandatory course for master study programmes Mathematical
analysis and Mathematical modelling in physics and technics. Recommended
for the first year of master studies. The course is devoted to advanced
topics in functional analysis - locally convex spaces and weak topologies,
theory of distributions, vector integration, compact convex sets.
Last update: Pyrih Pavel, doc. RNDr., CSc. (12.05.2022)
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The rules for 2022/2023:
The course is finished by a credit and an exam.
The credit will be awarded after complete and correct solution of one exercise, which is then presented by the student during an exercise lesson. Suitable exercises will be on the webpage of the lecturer (3 possible exercises for each week, available at least one week in advance). Booking the exercise is possible either personally, or by an email to cuth@....
Before passing the exam it is necessary to gain the credit. Last update: Kalenda Ondřej, prof. RNDr., Ph.D., DSc. (07.09.2023)
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Rudin, W.: Functional analysis. Second edition, McGraw-Hill, Inc., New York, 1991 (chapters 1-3 and 10-12)
M.Fabian et al.: Banach Space Theory, Springer 2011 (chapter 3)
J.Diestel and J.J.Uhl: Vector measures, Mathematical Surveys and Monongraphs 15, American Mathematical Society 1977 (sections III.1-III.3)
R.R.Ryan: Introduction to tensor products of Banach spaces, Springer 2002 (sections 2.3 and 3.3)
Meise R. and Vogt D. : Introduction to functional analysis, Oxford University Press, New York, 1997 (chapters 17 and 18) Last update: Kalenda Ondřej, prof. RNDr., Ph.D., DSc. (15.09.2023)
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The exam is oral with the possibility of a written preparation. Mainly knowledge and understanding of the notions and theorems explained during the semester will be tested. In addition, solving selected problems using the methods explained during the course will be a part of the exam. The lectures are the main source of materials for the exam. Last update: Cúth Marek, doc. Mgr., Ph.D. (29.09.2022)
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1. Locally convex spaces
Definitions of a topological vector spaces and of a locally convex space Minkowski functionals, seminorms, generating locally convex topologies using seminorms Boundedness in a locally convex space Metrizability and normability of locally convex spaces Continuous linear mappings between locally convex spaces, linear functionals Hahn-Banach theorem - extending and separating Fréchet spaces Weak topologies - topology generated by a subspace of the algebraic dual, weak and weak* topologies, Goldstine, Banach-Alaoglu, reflexivity and weak compactness, bipolar theorem 2. Elements of the theory of distributions Space of test functions and the convergence in it Distributions - definition, examples, operations, characterizations order of a distribution, convergence of distributions convolution of a distribution and a test function, approximate unit convolution of two distributions - examples that it sometimes works Schwarz space as a Fréchet space Tempered distributions and their characterizations Fouriera transform of tempered distributions convolution of tempered distributions possibuly the support of a distribution 3. Elements of vector integration Measurability of vector-valued functions, Pettis theorem Weak integrability, Dunford and Pettis integrals Bochner integral Bochner spaces Duality of Bochner spaces (briefly, no proofs) 4. Convex compact sets Extreme points Krein-Milman theorem integral representation theorem
Last update: Kalenda Ondřej, prof. RNDr., Ph.D., DSc. (09.05.2022)
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Mandatory course for master study branches Mathematical analysis and Mathematical modelling in physics and technics. It is required to know notions, methods and results from the course Introduction to Functional Analysis NMMA331. Knowledge of basic concepts of general topology (topological spaces, continuous mappings, compactness) is recommended. Last update: Kalenda Ondřej, prof. RNDr., Ph.D., DSc. (01.09.2021)
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