Module: Analysis 2

Latest Module Specifications

Current Academic Year 2025 - 2026

Module Title	Analysis 2
Module Code	MTH1043 (ITS: MS231)
Faculty	Mathematical Sciences	School	Science & Health
NFQ level	8	Credit Rating	7.5

Description

This module covers the essential parts of classical and modern Analysis as required for future courses on Probability and Statistics, Differential Equations, and Partial Differential Equations. The topics covered by this course include but are not restricted to: Metric spaces, Continuous functions, Convolutions, and Fourier Analysis

Learning Outcomes

1. Learn important mathematical concepts as used in subsequent courses in Probability Theory, Stochastic Processes, Differential Equations, and Partial Differential Equations
2. Apply theoretical concepts to particular modelling and computational problems related with real-world applications

*Type*	*Hours*	*Description*
Workload	Full time hours per semester
Lecture	36	Students will attend lectures where new material will be presented and explained. Also attention will be drawn to various supporting material and tutorials as the course progresses.
Tutorial	12	Students will show their solutions to homework questions and will receive help with and feed-back on these solutions.
Independent Study	78	Corresponding to each lecture students will devote approximately one additional hour of independent study to the material discussed in that lecture or to work on support material when attention is drawn to such in lectures. Before each tutorial students will devote approximately three and a half hours to solving homework problems which are to be discussed in that tutorial.
Total Workload: 126

Section Breakdown
CRN	11213	Part of Term	Semester 1
Coursework	0%	Examination Weight	0%
Grade Scale	40PASS	Pass Both Elements	Y
Resit Category	RC3	Best Mark	Y
Module Co-ordinator	Martin Friesen	Module Teacher

Type	Description	% of total	Assessment Date
Assessment Breakdown
Written Exam	n/a	20%	n/a
Formal Examination	n/a	80%	End-of-Semester

Reassessment Requirement Type
Resit arrangements are explained by the following categories; RC1: A resit is available for both^* components of the module. RC2: No resit is available for a 100% coursework module. RC3: No resit is available for the coursework component where there is a coursework and summative examination element. ^* ‘Both’ is used in the context of the module having a coursework/summative examination split; where the module is 100% coursework, there will also be a resit of the assessment

Pre-requisite	None
Co-requisite	None
Compatibles	None
Incompatibles	None

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

Metric and Normed spaces
metrics, norms, inner products, convergence, completeness, open and closed sets, compactness

Continuous functions
completeness, pointwise and uniform convergence, Arzela-Ascoli Theorem, Hoelder and Lipschitz continuity, Banachs fixed-point theorem, applications

Convolutions
elementary properties, Dirac approximation and mollifiers, (applications)

Harmonic Analysis
periodic functions, Fourier coefficients, representation by Fourier series, Plancherel identity, abstract Fourier decomposition in Hilbert spaces

Integration theory
Riemann-Stieltjes integration, integration of distribution functions, Young integral

Indicative Reading List

Books:
None

Articles:
None

Other Resources

None