Introduction to Functional Analysis, 5 op

The course provides an introduction to the theory of bounded linear operators on general vector spaces, with special emphasis on Banach and Hilbert spaces of functions.

List of the main topics:

• General linear vector spaces, normed spaces and inner product spaces. Completeness and topological aspects. Banach and Hilbert spaces.
• Vector spaces of functions.
  
• Bounded linear operators on Banach and Hilbert spaces.
  
• Hahn-Banach Theorem, Riesz Representation Theorem, Minimum Norm Theorem.
  
• Self-adjoint and compact operators on Hilbert spaces.
• Bases of infinite-dimensional vector spaces.
  
• 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.

The course in particular provides the mathematical background for the following further courses in mathematical analysis:

• MATH.MA.830 Advanced Functional Analysis
• MATH.APP.810 Mathematical Control Theory