Syllabus
*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
Annotation
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.