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Module Specifications

Current Academic Year 2012 - 2013
Please note that this information is subject to change.

Module Title	Modern Analysis
Module Code	MS301
School	School of Mathematics
Online Module Resources
Module Teacher	Niamh O'Sullivan
NFQ level	8	Credit Rating	7.5
Pre-requisite	None
Co-requisite	None
Compatibles	None
Incompatibles	None

Description

In this module students are introduced to the core subject of analysis that is required for other third and fourth year courses. The approach taken will encourage students to develop critical thinking and apply logical reasoning to mathemtical problems. Students will participate in the following learning activities: Lectures: Students will attend three one-hour lectures per week. These lectures are designed to introduce learners to the mathematical principles and problem solving techniques that underpin this module. Tutorials: Each student will attend one one-hour tutorial per week. Problem sheets based on lecture content are distributed to the students and they are strongly advised to attempt all tutorial questions in advance of the tutorial.Reading: Students are expected to fully utilise the textbooks recommended.

Learning Outcomes

1. Define metric space and associated properties, and recognise these properties in specific examples.
2. Interpret concepts from analysis of a single real variable (convergence, continuity) in the context of metric spaces.
3. Define open and closed sets, and demonstrate an understanding of how they relate to continuity, etc..
4. Explain important concepts such as compactness and completeness, recognise them in concrete examples, and use them to derive conclusions.

Workload	Full-time hours per semester
*Type*	*Hours*	*Description*
Lecture	36	3 lectures per week
Tutorial	12	1 tutorial per week
Independent learning	100	Study
Directed learning	24	Solving tutorial questions
Examination	2	End of semester exam
Total Workload: 174

All module information is indicative and subject to change. For further information,students are advised to refer to the University's Marks and Standards and Programme Specific Regulations at: http://www.dcu.ie/registry/examinations/index.shtml

Indicative Content and Learning Activities

Introduction.
Metric spaces, normed vector spaces, inner product spaces..

Continuity and Convergence.
Open and closed sets, limit points, convergence, Cauchy sequences, continuity, uniform continuity, completeness,..

Banach Contraction Mapping Principle.
Banach contraction mapping principle, applications to differential and integral equations, implicit function theorem..

Linear Operators.
Linear operators, bounded linear operators on normed spaces, norms defined on bounded linear operators, invertible linear operators, applications to integral equations..

Compact Spaces.
compact, sequentially compact and countably compact spaces, totally bounded spaces, maximum-minimum theorem..

Connectedness.
Connected spaces, path-connectedness, intermediate-value theorem..

Assessment Breakdown
Continuous Assessment	25%	Examination Weight	75%

Course Work Breakdown
Type	Description	% of total	Assessment Date
Short answer questions	Class Test	10%	Week 4
Short answer questions	Class Test	10%	Week 8
Short answer questions	Class Test	10%	Week 12

Reassessment Requirement
Resit arrangements are explained by the following categories; 1 = A resit is available for all components of the module 2 = No resit is available for 100% continuous assessment module 3 = No resit is available for the continuous assessment component
This module is category 3

Indicative Reading List

Victor Bryant: 1985, Metric Spaces, Cambridge University Press,
E. Kreyzig: 1989, Introductory Functional Analysis with Applications, Wiley,
Seymour Lipschutz: 0, General Topology (Schaum's outlines),, McGraw-Hill,

Other Resources

None

Array

Programme or List of Programmes

APM

B.Sc. Applicable Mathematics

BSc in Financial & Actuarial Mathematics

Timetable this semester: Timetable for MS301

Date of Last Revision

11-OCT-10

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