Aim of the course
The aim of this course is to obtain a broad basis in functional analysis well beyond the introductory level. Specifically, the goals of the course are that
- the student acquires a solid understanding of the basic results and techniques in linear functional analysis.
- the student understands the spectral theorem for (unbounded) normal operators as it is derived in the context of C*-algebras.
- the student understands the applications of functional analysis to for instance measure theory, Fourier theory, differential equations.
We start with a quick introduction to Banach and Hilbert space theory (recalling the material covered in introductory bachelor courses on Functional Analysis). Then, we discuss weak topologies in the framework of locally convex spaces. We then change our perspective to discuss spaces of operators on these Hilbert spaces, leading us to the theory of C*-algebras and eventually to the spectral theorem of (bounded) normal operators. After a careful discussion of unbounded symmetric and self-adjoint operators, we prove the spectral theorem for unbounded normal operators. Finally, we come to the basics of the Fredholm index.