An Introduction to Hilbert Spaces
Not available in 2013
Previously offered in 2012, 2011, 2010, 2009, 2008, 2007, 2006, 2005, 2004
Introduces the most user-friendly of the various spaces occurring in analysis, chosen because their geometry is most like that of familiar two and three dimensional Euclidean space. Hilbert Spaces provide an excellent framework for the study of quantum mechanics, classical subjects like Fourier analysis and the developing theory of wavelets. Consequently they arise frequently in application such as theoretical physics, control theory and image processing. They also underlie much of the research effort of mathematicians in the Functional Analysis Group at Newcastle.
|Objectives||On successful completion of this course, students will have:
1. an awareness of the breadth of mathematics as well as an in depth knowledge of one specific area.
2. an ability to communicate a convincing and reasoned argument of a mathematical nature in both written and oral form.
3. an understanding of what constitutes a rigorous mathematical argument and how to use reasoning effectively to solve problems.
|Content||* Inner product spaces.
* Review of finite-dimensional inner product spaces, infinite dimensional examples.
* The inner product norm, Cauchy-Schwarz inequality, orthogonality, and Pythagoras' theorem.
Convergent sequences and series, Cauchy sequences and completeness.
* Closed subspaces, best approximations, projections and the orthogonal decomposition of vectors.
* Orthonormal sets, Bessel's inequality, least square best approximation, complete orthonormal sets, Parseval's identity.
* Fourier series and their convergence.
* Linear operators on Hilbert space: continuity, Riesz representation theorem and ajoint operators.
* Self-adjoint operators, eigenfunction expansions.
|Assumed Knowledge||MATH2320 and MATH2330|
|Modes of Delivery||Internal Mode|
|Contact Hours||Lecture: for 3 hour(s) per Week for Full Term|