Introduction to Functional Analysis, 5 op

Core content

• Introduction to metric spaces and their fundamental properties, especially spaces of continuous and integrable functions. Analysis of continuous mappings between two metric spaces. Cauchy-sequences, complete metric spaces and completions. The Banach-Steinhaus fixed point theorem.
• General linear vector spaces and normed spaces. Basics of Banach spaces and the theory of linear operators. Analysis of properties differential equations using operator theory.
• Fundamentals of Hilbert spaces. Operator theory in Hilbert spaces, bounded self-adjoint operators. Minimum norm theorem and the Riesz reperesentation theorem.
• Spectral theory of bounded linear operators. The spectral representation of compact self-adjoint operators.
• Analysis of solvability and solutions of integral equations using operator theory and spectral theory.