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Semester 1

Linear Analysis (MATH10082)

Subject

Mathematics

College

SCE

Credits

10

Normal Year Taken

4

Delivery Session Year

2022/2023

Pre-requisites

Visiting students are advised to check that they have studied the material covered in the syllabus of each prerequisite course before enrolling.

Course Summary

In this course, we will introduce students to techniques and tools in modern analysis which have important uses in a variety of areas of analysis, including the study of partial differential equations and Fourier analysis.We will achieve this in the context of linear analysis, introducing normed linear, inner product spaces and their completions, Banach and Hilbert spaces. The structure and geometry of these spaces will be studied as well as continuous linear operators acting on them. Many examples will be studied as well as connections to other fields.

Course Description

-Inner product spaces and normed spaces.-Completeness and completions of spaces with concrete realisations of standard examples. Lp spaces, Holder and Minkowski inequalities.-Geometric and metric properties of Hilbert spaces, including orthonormal bases and generalised Fourier series.-Bounded linear functionals, operators and duality,-Spectral Theorem for compact, self-adjoint operators on a Hilbert space.

Assessment Information

Written Exam 95%, Coursework 5%, Practical Exam 0%

Additional Assessment Information

Coursework 20%, Examination 80%

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