2017/18 Taught Postgraduate Module Catalogue

MATH5015M Linear Analysis 1

20 creditsClass Size: 20

Module manager: Dr Vladimir Kisil
Email: V.Kisil@leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2017/18

Pre-requisite qualifications

MATH3215, or equivalent.

This module is approved as an Elective

Module summary

Areas and volumes of geometrical figures are notions which we regularly meet in our practical activities. We learn formulae for areas of simple shapes in elementary geometry. However a consistent extension of that notion to much more complicated sets is not a trivial matter. This is the subject of measure theory, which is a foundation for the theory of integration. The construction essentially uses tools of linear analysis: the duality of vector spaces, a continuation of a function from a dense set in a continuous manner, etc. Overall, the module provides a representative sample of useful methods employed in modern analysis.

Objectives

On completion of this module, students should be able to:

a) solve problems concerning Lebesgue measure;
b) solve problems about Banach spaces;
c) apply these ideas to Fourier Series.

Syllabus

1) Normed spaces, bounded linear operators on a Banach space, dual spaces, Hahn-Banach theorem, Zorn's lemma. Use of sequence spaces. The Banach space C(X) for a compact space X.
2) Basic measure theory, up to the construction of the Lebesgue measure on the real line. Complex measures and measurable functions. Dominated convergence theorem. Product measures. Fubini theorem.
3) Definition of spaces of Lebesgue integrable functions, and proof that with the standard norm they form a Banach space. Dual spaces. The Radon-Nikodym Theorem. The conjugate index theorem.
4) The Banach space M(X) of regular Borel measures on a compact space X. Proof that the dual of C(X) is M(X).
5) Applications to Fourier series.

Teaching methods

Delivery type	Number	Length hours	Student hours
Lecture	44	1.00	44.00
Private study hours			156.00
Total Contact hours			44.00
Total hours (100hr per 10 credits)			200.00

Private study

Studying and revising of course material.
Completing of assignments and assessments.

Opportunities for Formative Feedback

Regular problems sheets

Methods of assessment

Exams

Exam type	Exam duration	% of formal assessment
Standard exam (closed essays, MCQs etc)	3 hr 00 mins	100.00
Total percentage (Assessment Exams)		100.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

Reading list

The reading list is available from the Library website

Last updated: 26/04/2017

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