CloseHelpPrint
Kies de Nederlandse taal
Course module: 191506302
191506302
Applied Functional Analysis
Course infoSchedule
Course module191506302
Credits (ECTS)6
Course typeCourse
Language of instructionEnglish
Contact personprof.dr. C. Brune
E-mailc.brune@utwente.nl
Lecturer(s)
Examiner
prof.dr. C. Brune
Contactperson for the course
prof.dr. C. Brune
Examiner
dr. G.A.M. Jeurnink
Lecturer
dr. F.L. Schwenninger
Academic year2018
Starting block
1A
Application procedureYou apply via OSIRIS Student
Registration using OSIRISYes
Aims
Learning goals
In contemporary mathematics (infinite dimensional) 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 how this feature of applied mathematics is operating, that’s the main goal for the student to learn functional analysis
Content
Course description
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.
Prior knowledge
Standard bachelor courses in Linear Algebra and real and complex Analysis. 
Assumed previous knowledge
Gewenst: Knowledge of Calculus and Linear Algebra
Participating study
Master Applied Mathematics
Required materials
Syllabus
-
Recommended materials
-
Instructional modes
Lecture

Tutorial

Tests
Written exam

CloseHelpPrint
Kies de Nederlandse taal