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M4062 Banach algebras
Aims:
The main aim of this unit is to present a number of major theorems in functional analysis. This branch of mathematics is fundamental to the study of various areas of mathematical physics, such as quantum mechanics and to the modern theory of partial differential equations. The unit progresses from Hilbert spaces, through Banach spaces and Banach algebras to C*-algebras, and provides a background to reading in operator theory. In addition it provides some startling applications of the theory of complex analysis.
Syllabus:
Banach spaces. Linear transformations. The closed graph and open mapping theorems. Dual spaces. Hahn-Banach theorem. Alaoglu theorem. Banach algebras. Maximal group. Spectrum and resolvent. Gelfand-Mazur theorem. Maximal ideals of commutative Banach algebras. Gelfand representation theorem. Stone-Weierstrass theorem. C*-algebras. Gelfand-Naimark representation theorem.
Prerequisites:
MTH3132 Analysis and Geometry;
MTH3021 Dynamical Systems and Complex Analysis;
M4052 Topology is also recommended as a co-requisite.
References:
Simmons, G.F., Introduction to topology and modern analysis, McGraw-Hill.
Lecturer: Alan Pryde
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