In contemporary mathematics, Functional Analysis plays an important role. Concepts such as Hilbert space, and in general normed space, and operators between these spaces are nowadays common tools. In many applications from Mathematical Physics to Financial Engineering, from Computational Mechanics to Statistics, this generally developed theory is applicable.
The understanding of a clean abstract theory as an indispensable tool for application within mathematics, that’s the main goal for the student to learn functional analysis
The main subject of Functional Analysis is the analytic theory of infinite-dimensional linear spaces. The language is the language of linear algebra and geometry, but the subtleties of the subject are in analysis. This is for instance shown by the fact that ‘norm completeness’ is essential in most of the deeper theorems: Fourier series, best approximations in Hilbert spaces, series in a Banach space, Riesz theorem, spectral theory. The insight provided by the abstract point of view is of astonishing beauty; it makes the formulas, derived in classical courses, comprehensible.
The heart of the course is the theory of operators (maps acting on a class of functions, with special attention to functionals). These can be seen as higher level objects: for instance, typical operators considered in this course map sequences (or functions) to sequences (or functions). In terms of linear algebra, this means that vectors are replaced by functions and the substitute for (linear mappings represented by) matrices are operators.
The course starts with the introduction of normed linear spaces (including Lebesque spaces). The special class of Hilbert spaces allows for some additional structure in which we can solve approximation problems in optimization theory. Moreover, this structure also allows for a rigorous characterization (Riesz representation theorem) of the functionals acting on Hilbert spaces: the dual space. As an application this allows for an abstract framework to treat the Galerkin method for boundary-value problems. We also examine the spectrum of a bounded linear operator, especially the one which belongs to a self-adjoint compact operator and covered by the spectral theorem. We briefly also touch fundamental theorems of functional analysis such as Baire's theorem and the Hahn-Banach theorem. Optionally, the topic of unbounded operators in order to tackle differential equations, will be treated.
The lectures of this master course give the student the possibility to solve exercises in a classroom setting with assistance, as well as to deliver these as homework for bonus. A sufficient mark for the oral examination concludes this course for the student.
Standard math bachelor courses in Linear Algebra and real (and complex) Analysis. Note that we will heavily rely on concepts from Analysis I and II (continuous functions, convergence of sequences of functions, compact sets, etc).