Code: 2-MPG-105
Continuous assessment: None
Final assessment: Examination with problems
Objective: To provide basic education in general topology and functional analysis for non-specialists for use in applications.
Course syllabus:- Topological space. Metric topology.
- Open and closed sets, neighbourhoods. Interior and closure.
- Continuous mapping.
- Basic constructions of topological spaces (subspace, finite product, quotient space).
- Countability axioms. Separation (T1 - T4). Connectedness. Compactness.
- Linear normed spaces, linear continuous functionals and operators, Hahn-Banach theorem, dual spaces, Banach spaces, Banach-Steinhaus theorem, differences between finite-dimensional and infinite-dimensional spaces.
Hilbert spaces, projection theorem, Riesz representation theorem, Bessel's inequality, Fourier coefficients, orthogonal systems, orthonormal basis. - Spaces of continuous functions, Stone-Weierstrass theorem, dual of the space C(I).
Literature:
Kelley, J.: General Topology, 1957.
Rudin, W.: Real and Complex Analysis, Academia 1977.
Kolmogorov, A., Fomin, S.: Foundations of the Theory of Functions and Functional Analysis, 1975.
Taylor, A.: Introduction to Functional Analysis, Academia 1973.
The lecture follows the text Topology and Functional Analysis. Exercises can also be found there.
Rudin, W.: Real and Complex Analysis, Academia 1977.
Kolmogorov, A., Fomin, S.: Foundations of the Theory of Functions and Functional Analysis, 1975.
Taylor, A.: Introduction to Functional Analysis, Academia 1973.
The lecture follows the text Topology and Functional Analysis. Exercises can also be found there.