2016/17 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 2016/17
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 sheetsMethods 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 websiteLast updated: 08/04/2016
Browse Other Catalogues
- Undergraduate module catalogue
- Taught Postgraduate module catalogue
- Undergraduate programme catalogue
- Taught Postgraduate programme catalogue
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