MATH40270 Analysis for Graduates

Academic Year 2012/2013

This course will provide students with the fundamentals of modern analysis. Among the topics covered will be the following.TOPOLOGY: Review of Set Theory, Ordering, Equivalence Relations, Point Set Topology, Connected Sets, Compact Sets MEASURE THEORY: Lebesgue Measure, Measure Spaces, Measurable Functions, Integration, Convergence Theorems, Riesz Representation Theory FUNCTION ANALYSIS: Review of conditional, absolute and uniform convergence, Banach Spaces, Hilbert Spaces, Projections on Hilbert Spaces, l_p, L_p, C(K) spaces, Baire category, Linear functionals, Duality, Reflexivity, Weak and weak-star topologies, Hahn-Banach Theorem, OPERATOR THEORY: Linear Operators, Compact Operators, Spectral Theory HILBERT SPACES: Examples of Prehilbert and Hilbert spaces, projections, representations of linear functionals, BANACH ALGEBRAS: Definitions, Ideals, Quotient Algebras

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Curricular information is subject to change

Learning Outcomes:

On successful completion of the course the student should 1) have a wide knowlegde and deep understanding of the fundamental of modern analysis 2) be parpared for the study more specialised graduate courses in algebra, analysis and geometry 3) have the confidence to analyse and solve non trivial problems in analysis 4) appreciate the interconnection with varies aspect of mathematics

Student Effort Hours: 
Student Effort Type Hours
Lectures

48
Tutorial

12
Autonomous Student Learning

240
Total

300
 
Requirements, Exclusions and Recommendations
Learning Recommendations:

Student be familar with the basics of real and compex analysis and of linear algebra. A basic knowledge of metric spaces would also be very useful.



 
Description % of Final Grade Timing
Examination: < Description >

70

 3 hour End of Trimester Exam
Examination: < Description >

30

 Unspecified

Compensation

This module is passable by compensation

Resit Opportunities

End of Semester Exam

Remediation

If you fail this module you may repeat, resit or substitute where permissible