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This is a mandatory course for the master’s study programmes Mathematical Modelling in
Physics and Technology, which can be substituted by Functional Analysis 1. It is recommended
for the first year of master’s studies. The course focuses on advanced topics in measure theory
and functional analysis.
Last update: Šmíd Dalibor, Mgr., Ph.D. (22.05.2025)
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The course is concluded with a credit and a final exam. Obtaining the credit is a prerequisite for taking the exam. The credit will be granted upon the complete and correct solution of the assigned homework. Detailed requirements will be provided on the lecturer’s webpage. Last update: Šmíd Dalibor, Mgr., Ph.D. (22.05.2025)
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W. Rudin: Functional analysis. Second edition, McGraw-Hill, Inc., New York, 1991 M. Fabian et al.: Banach Space Theory, Springer 2011 J. Diestel and J. J. Uhl: Vector measures, Mathematical Surveys and Monongraphs 15, American Mathematical Society 1977 R. R. Ryan: Introduction to tensor products of Banach spaces, Springer 2002 J. Lukeš and J. Malý: Measure and integral. MatfyzPress, Charles University, Prague, 1995. Last update: Šmíd Dalibor, Mgr., Ph.D. (22.05.2025)
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The exam is oral, with the option for written preparation. It will primarily test knowledge and understanding of the concepts and theorems covered during the semester. Additionally, the exam will include solving selected problems using the methods presented in the course. The lectures serve as the main source of material for the exam. Last update: Šmíd Dalibor, Mgr., Ph.D. (22.05.2025)
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1) Elements of Topology: Basic concepts including topological spaces, continuity, compactness.
2) Convex Sets and duality: Convex sets, convex hull and its properties, Legendre transforms, convex minimisation theorems.
3) Fixed Point Theorems: Fundamental principles and their applications in nonlinear analysis.
4) Measure Spaces and σ-Algebras: Foundations and fundamental concepts of measure theory.
5) Classical Approximation Theorems: Lusin’s, Egorov’s, and Kolmogorov’s theorems in measure theory.
6) Radon Measures: Definition, essential properties, and characterization; duality of spaces of continuous functions.
7) Radon-Nikodym Property: Statement, proofs, and applications in analysis.
8) Dual Spaces of L^p: Structural analysis and representation theorems.
9) Vector Integration: Introduction to the Bochner integral and its key properties.
10) Space L^p(X): Characterization of the dual spaces, weakly compact sets in L^1(X) Last update: Šmíd Dalibor, Mgr., Ph.D. (22.05.2025)
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Fundamentals of Banach and Hilbert spaces, including elements of Fredholm theory, spectral theory of bounded operators, and properties of compact and self-adjoint operators. Understanding of distributions, the Lebesgue integral and measure, as well as the Stokes theorem. Last update: Šmíd Dalibor, Mgr., Ph.D. (22.05.2025)
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