The aim of the course is to acquaint students with basic properties of Banach and Hilbert spaces as well as with basis of the theory of linear operators on Banach spaces.
Wymagania wstępne
It is assumed that students know basis of set theory, metric topology, linear algebra, mathematical analysis and elements of measure and Lebesgue integral theories.
Zakres tematyczny
Lecture
Metric and Frechet spaces
Definition of a metric space, verification if a given function is a metric, compact sets in metric spaces,
Theorem of the continuity of functions in metric spaces .
Weaker, stronger and equivalent metrics (definitions and examples).
Definition of an F-norm and a Frechet space.
Theorem about relationships between metrics and F-norms.
Normed and Banach spaces
Definition of a norm and a Banach space, verification if a given function is a norm.
Classical function Banach spaces C[a,b], Lp[a,b] 1≤p<∞, L∞[a,b], definitions and propeties.
Minkowski inequalities for sums and integrals.
Containing of Lp[a,b] spaces as sets.
Hilbert spaces
Definitions of an inner product, a unitary and a Hilbert space (examples).
Theorem of the Schwarz inequality.
Two theorems about relationships between a norm and an inner product.
Theorem of the parallelogram identity.
Theorem of the independency of elements of an orthogonal system.
Schmidt theorem of the orthogonality of the system (applicability of the algorithm).
Bounded linear operators on normed spaces
Linear operators in Banach spaces (investigation of the linearity in examples).
Two theorems about continuity of linear operators in Banach spaces.
“The open mapping” and “ the closed graph” theorems.
The norm of a bounded linear operator (definition and investigation).
Banach-Steinhaus theorem and its applications.
Hahn-Banach theorem and its applications .
General form of continuous linear functionals over classical Banach spaces.
Classes
Normed and Banach spaces
Examples of sequence and function linear spaces. Basic properties. Hölder and Minkowski inequalities.
Examining norm conditions on sequence and function spaces. Proving completeness of classical sequence and function normed spaces.
Calculation of the norm of elements in sequence and function spaces.
Comparing norms in normed spaces.
Hilbert spaces
Examples of Hilbert. Basic properties.
Examining conditions of inner product in sequence and function spaces.
Examining orthogonal systems in Hilbert spaces.
Bounded linear operators on normed spaces
Examining linearity and boundedness of functionals and operators defined on sequence and function normed spaces.
Calculation of the norm of linear functionals on sequence and function spaces.
Colloquium
Metody kształcenia
Conventional (traditional) lecture. Classes (auditorium), solving exercises and problems.
Efekty uczenia się i metody weryfikacji osiągania efektów uczenia się
Opis efektu
Symbole efektów
Metody weryfikacji
Forma zajęć
Warunki zaliczenia
The course completion grade consists of the classes grades (40%) and examination grades (60%). A positive grade of the classes is required to sit for the exam. A positive grade of the examination is required to credit for the course.
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