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 made clear 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. This is why we are focusing on concepts like ‘completeness’ and ‘denseness’.
As mentioned, the theory of Fourier series and best approximations is part of our investigations. 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). Be aware that this is a big mental step: we could call them “third level objects”, after points in Rk (level 1) and functions on Rk (level 2). Here we encounter differential and integral operators: historically spoken is calculus of variation the source of Functional Analysis. We incorporate these examples and applications .
The course starts with the introduction of normed linear spaces (including Lebesque spaces). We first restrict ourselves to Hilbert spaces, a category in which we can solve approximation problems in optimization theory. A crucial role is played by operators between spaces (of functions). By the Riesz theorem we can describe the dual of a Hilbert space and treat the Galerkin method for boundary-value problems. We also examine the spectrum of a bounded linear operator, especially the one who belongs to a self-adjoint compact operator. This enables us to tackle Sturm-Liouville problems. We also pay attention to unbounded operators in order to tackle differential equations.
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 written examination concludes this course for the student.
Standard bachelor courses in Linear Algebra and real and complex Analysis.