Introduction to Linear Analysis (MAGIC003) |
GeneralDescription
This couse provides an introduction to analysis in infinite dimensions with a minimum of prerequisites. The core of the course concerns operators on a Hilbert space including the continuous functional calculus for bounded selfadjoint operators. There will be an emphasis on positivity and on matrices of operators.
The course includes some basic introductory material on Banach spaces and Banach algebras. It also includes some elementary (infinite dimensional) linear algebra that is usually excluded from undergraduate curricula.
Here is a very brief list of the many further topics that this course looks forward to.
Banach space theory and Banach algebras; C^{*}-algebras, von Neumann algebras and operator spaces (which may be viewed respectively as noncommutative topology, noncommutative measure theory and `quantised' functional analysis); Hilbert C^{*}-modules; noncommutative probability (e.g. free probability), the theory of quantum computing, dilation theory;Unbounded Hilbert space operators, one-parameter semigroups and Schrodinger operators. And that is without starting to mention Applied Maths and Statistics applications ...
Relevant books
SemesterAutumn 2009 (Monday, October 5 to Friday, December 11) Timetable
Prerequisites
Standard undergraduate linear algebra and real and complex analysis, and basic metric space/norm topology.
Syllabus
I Preliminaries (5 lectures)
Students
Bibliography
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