MATH3050 Functional Analysis (5 cr)
Cooperation network course
Network: Vaasa Higher Education Consortium
Network: Cross-institutional studies in advanced courses in mathematics and statistics
This course is offered through the Network for Advanced Studies in Mathematics. These studies are available for the following degree students:
- Bachelor's Degree Programme in Mathematics
- Master's Degree Programme in Mathematics
- Bachelor's Degree Programme in Mathematics (Subject Teacher)
- Master's Degree Programme in Mathematics (Subject Teacher)
- Bachelor's Degree Programme in Mathematics, Chemistry or Physics Subject Teacher Education and Primary Teacher Education (Specialication in Mathematics)
- Master's Degree Programme in Mathematics, Chemistry or Physics Subject Teacher Education and Primary Teacher Education (Specialication in Mathematics)
- Doctoral Programme in Mathematics and Statistics
- Doctoral Programme in Mathematics and Science (Specialication in Mathematics)
Description
Linear spaces, linear operators, quotient spaces, dual spaces. Normed linear spaces, some topological notions, inner product spaces, orthonormal systems and the Gram-Schmidt procedure. Banach spaces and Hilbert spaces, separable spaces, orthonormal bases in Hilbert spaces, Fourier series. Bounded linear operators in Hilbert spaces, Riesz representation theorem, adjoint operator, orthogonal projectors, normal, self-adjoint and compact operators on Hilbert spaces, Fredholm and Volterra integral operators. Open mapping theorem, closed graph theorem, and uniform boundedness principle. Dual spaces and the Hahn-Banach theorem, reflexive spaces, weak convergence and weak∗ convergence, Sobolev spaces.
Learning outcomes
Students learn the fundamentals of linear spaces, particularly infinite-dimensional vector spaces, normed linear spaces, inner product spaces, and their completeness. In particular, students will become familiar with Banach and Hilbert spaces, such as sequence and function spaces, as well as linear operators acting on such spaces. The course provides students with a general understanding of the fundamental theorems of functional analysis, their important consequences, and useful applications, covering topics in operator and spectral theory. Functional analytic methods and operator and spectral theoretic tools are widely used in different areas of scientific research.
Description of prerequisites
Basic courses in calculus and linear algebra.