Undergraduate - UnitMTH3160 - Analysis and topology

This unit entry is for students who completed this unit in 2014 only. For students planning to study the unit, please refer to the unit indexes in the the current edition of the Handbook. If you have any queries contact the managing faculty for your course or area of study.

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6 points, SCA Band 2, 0.125 EFTSL

Refer to the specific census and withdrawal dates for the semester(s) in which this unit is offered, or view unit timetables.

Level	Undergraduate
Faculty	Faculty of Science
Organisational Unit	School of Mathematical Sciences
Offered	Clayton First semester 2014 (Day)
Coordinator(s)	Dr John Head

Synopsis

In this unit we develop the theory of Banach spaces. Topics covered include a basic introduction to normed spaces, topology in Banach spaces, dual spaces, continuous linear mappings between Banach spaces, weak convergence and weak compactness in separable Banach spaces, Hilbert spaces and the Riesz representation theorem. Applications of these theories may include the contraction mapping theorem and its usage to prove the Cauchy-Lipschitz theorem (existence and uniqueness of solution to ordinary differential equations).

Outcomes

On completion of this unit students will be able to:

Understand the basic topological properties of normed spaces, and their applications to problems in other areas of mathematics;

Understand and appreciate some important basic theorems in analysis and their applications, such as the contracion mapping theorem and the Riesz representation theorem;

Recognise the conditions for existence and uniqueness of solutions to the initial value problem for systems of ordinary differential equations;

Communicate mathematical ideas and work in teams as appropriate for the discipline of mathematics.

Assessment

Projects (two): 20%
Weekly assignments (10): 10%
Final examination (three hours): 70%

Chief examiner(s)

Dr Jerome Droniou

Workload requirements

Three 1-hour lectures and one 1-hour support class per week

This unit applies to the following area(s) of study

Mathematics and statistics

Prerequisites

MTH2021 (with MTH2140 or MTH3140 as an additional prerequisite from 2015)

Co-requisites

MTH2140 or MTH3140 (for 2014 only)