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University of Wales, Bangor - School of InformaticsMathematics H2 Modules |
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Lecturer: Dr Gareth Roberts
Additional Reading:
Aims & Objectives:
To understand a variety of basic principles of functional analysis.
Prove and apply with confidence elementary results in functional analysis.
Pre-requisites: IAL2035, IAN2012.
Syllabus
Preliminaries of analysis and linear algebra:
Vector spaces, Hamel basis, dimension, linear transformations,
linear functionals, algebraic dual spaces, Euclidean spaces.
Continuous and differentiable functions on R, C, C^n, C^{\infty},
Lipshitz and Holder spaces.
Introduction to topological, metric and measure spaces:
Lebewsgue measure and integration: Lp spaces.
Continuity and compactness, compelteness, completion,
contraction mappings and fixed points.
Banach spaces:
Basic theory of normed vector spaces,
bounded linear operators and functionals.
the Hahn-Banach extension theorem, topological duals, weak convergence.
Hilbert spaces:
Basic theory of inner product spaces, orthogonal projections,
orthonormal bases, Fourier series.
Duality, Riesz representation theorem, adjoint operators.
Spectral theory, resolvent set and spectrum,
spectral theory for continuous, compact and self-adjoint operators.
Sobolev spaces:
An introduction to the theory of distributions
and weak (distributional) derivatives.
Assessment
2 hour end of semester closed book examination: 100%
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