This module on functional analysis aims to study normed linear spaces and some of the linear operators between them and give some applications of their use. The normed linear spaces which are complete metric spaces are especially important and the first part of the course is an introduction to the theory of Lebesgue integration with the aim of providing examples of complete spaces normed linear spaces of integrable functions.
1.1 σ algebras, measures and measurable functions:σ algebras, measures, measurable functions, characteristic functions, simple measurable functions, approximation of positive measurable functions by simple measurable functions., 1.2 , 1.3 Integration:The integral of simple positive measurable functions, positive measurable functions and measurable functions, the monotone convergence theorem, Fatou’s lemma and the dominated convergence theorem, the spaces of functionsL1, L2andL∞., 1.4 , 1.5 Normed spaces and inner product spaces:Cauchy Sequences, Complete spaces, Normed spaces, Banach spaces, inner product spaces, Hilbert spaces, Cauchy Schwarz inequality, orthogonality, Pythagoras theorem, orthogonal complement., 1.6 , 1.7 Linear operators: Continuous linear transformations, norm of continuous linear transformations, properties of the space BX, Y, the Riesz-Frechet theorem., 1.8 , 1.9 Operators on Hilbert spaces:The adjoint of an operator, normal, self-adjoint and unitary operators.
By the end of the course, students should be able to do the following:
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SCQF Level: 10
Credits: 15