In the first we will introduce Hilbert spaces via a number of canonical examples, and investigate the geometric parallels with Euclidean spaces (inner product, expansion in terms of basis elements, etc.). The module falls, roughly, into three parts. The fact that the spaces are infinite-dimensional introduces new possibilities, and much of the theory is devoted to reasserting control over these under suitable conditions. Hilbert spaces retain many of the familiar properties of finite-dimensional Euclidean spaces ( ) - in particular the inner product and the derived notions of length and distance - while requiring an infinite number of basis elements. They also form the cornerstone of the modern theory of partial differential equations.
#Functional analysis series
The ideas presented here allow for a rigorous understanding of Fourier series and more generally the theory of Sturm-Liouville boundary value problems. MA4A2 Advanced Partial Differential EquationsĬontent: This is essentially a module about infinite-dimensional Hilbert spaces, which arise naturally in many areas of applied mathematics.MA4A7 Quantum Mechanics: Basic Principles and Probabilistic Methods.Leads To: The following modules have this module listed as assumed knowledge or useful background: MA260 Norms, Metrics, & Topologies or MA222 Metric Spaces.MA244 Analysis III or MA258 Mathematical Analysis III.
0 kommentar(er)
