618-302 Linear Analysis

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Objectives:

On completion of this subject, students should:

Comprehend:

• fundamental concepts about measures and Lebesgue integration with respect to a variety of measures;

• how basic concepts of linear algebra can be generalised to infinite dimensional situations using techniques from analysis;

• how these concepts arise in many branches of mathematics, as for example, in partial differential equations, operations research and probability, but also in areas of theoretical physics such as quantum mechanics.

Have developed:

• the ability to give rigorous mathematical arguments at an advanced level;

• the ability to apply abstract concepts to solving problems in other areas of mathematics such as differential equations.

Appreciate:

• the necessity for a rigorous theoretical foundation of concepts used frequently in mathematics and physics.

Content:

Linear spaces and operators: Normed and inner product spaces, Hilbert spaces, abstract Fourier series; linear functionals and operators; dual spaces. Measure and integration: Introduction to measure and integration; dominated convergence and applications.

Assessment: